Step 1: Identifying the Derivative Relationship

To find the anti-derivative (integral) of the function $f(x) = \cos 3x$ by the method of inspection, we must identify a function $F(x)$ such that its derivative yields the original function:

$\frac{d}{dx} [F(x)] = \cos 3x$ [Definition of Anti-derivative]

Step 2: Selecting the Trial Function

We know from the standard trigonometric derivatives that the derivative of the sine function results in a cosine function:

$\frac{d}{dx} (\sin ax) = a \cos ax$ [Chain Rule of Differentiation]

In our case, $a = 3$. Therefore, let us consider the function $\sin 3x$.

Step 3: Calculating the Derivative of the Trial Function

Differentiating $\sin 3x$ with respect to $x$:

$\frac{d}{dx} (\sin 3x) = \cos 3x \cdot \frac{d}{dx} (3x)$ [Applying the Chain Rule]

$\frac{d}{dx} (\sin 3x) = \cos 3x \cdot 3$

$\frac{d}{dx} (\sin 3x) = 3 \cos 3x$

Step 4: Isolating the Target Function

To isolate $\cos 3x$ as requested by the inspection method, divide both sides of the equation by $3$:

$\frac{1}{3} \cdot \frac{d}{dx} (\sin 3x) = \cos 3x$

Using the linearity property of derivatives, $\frac{d}{dx} [k \cdot f(x)] = k \cdot \frac{d}{dx} [f(x)]$, we can move the constant inside the derivative operator:

$\frac{d}{dx} \left( \frac{1}{3} \sin 3x \right) = \cos 3x$

Step 5: Visualizing the Relationship

Final Solution: The anti-derivative of $\cos 3x$ is $\frac{1}{3} \sin 3x$.

Step 1: Algebraic Simplification of the Integrand

Given the integral: $I = \int (1 - x)\sqrt{x} \, dx$

To facilitate integration, we apply the distributive property of multiplication over subtraction [By the Distributive Law: $a(b-c) = ab - ac$]. First, express the radical as a fractional exponent: $\sqrt{x} = x^{1/2}$.

$I = \int (1 - x) \cdot x^{1/2} \, dx$

$I = \int (x^{1/2} - x^1 \cdot x^{1/2}) \, dx$

Using the Laws of Exponents ($x^a \cdot x^b = x^{a+b}$), we compute the power for the second term:

$I = \int (x^{1/2} - x^{1 + 1/2}) \, dx$

$I = \int (x^{1/2} - x^{3/2}) \, dx$

Step 2: Application of Linearity and the Power Rule

By the linearity property of integration [$\int [f(x) \pm g(x)] \, dx = \int f(x) \, dx \pm \int g(x) \, dx$], we split the integral:

$I = \int x^{1/2} \, dx - \int x^{3/2} \, dx$

Apply the Power Rule for integration, which states $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ for $n \neq -1$:

$I = \frac{x^{(1/2) + 1}}{(1/2) + 1} - \frac{x^{(3/2) + 1}}{(3/2) + 1} + C$

Step 3: Arithmetic Simplification

Calculate the exponents and the denominators:

For the first term: $\frac{1}{2} + 1 = \frac{3}{2}$

For the second term: $\frac{3}{2} + 1 = \frac{5}{2}$

$I = \frac{x^{3/2}}{3/2} - \frac{x^{5/2}}{5/2} + C$

Multiply by the reciprocal of the fractions:

$I = \frac{2}{3}x^{3/2} - \frac{2}{5}x^{5/2} + C$

Visualization of the Function

Final Solution: $\frac{2}{3}x^{3/2} - \frac{2}{5}x^{5/2} + C$

Step 1: Application of the Linearity Property of Integrals

The integral of a sum of functions is equal to the sum of the integrals of the individual functions. Given the integral $I = \int (2x - 3\cos x + e^x) dx$, we apply the linearity property [Per the linearity property of integration $\int [af(x) + bg(x)] dx = a\int f(x) dx + b\int g(x) dx$]:

$I = \int 2x \, dx - \int 3\cos x \, dx + \int e^x \, dx$

$I = 2 \int x \, dx - 3 \int \cos x \, dx + \int e^x \, dx$

Step 2: Integration of Individual Terms

We evaluate each integral using standard integration formulas:

• For the first term, we apply the power rule $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$:
  $\int x^1 \, dx = \frac{x^{1+1}}{1+1} = \frac{x^2}{2}$
• For the second term, we use the standard trigonometric integral $\int \cos x \, dx = \sin x + C$:
  $\int \cos x \, dx = \sin x$
• For the third term, we use the exponential integral rule $\int e^x \, dx = e^x + C$:
  $\int e^x \, dx = e^x$

Step 3: Synthesis and Integration Constant

Substituting the results from Step 2 into the expression derived in Step 1:

$I = 2 \left( \frac{x^2}{2} \right) - 3(\sin x) + e^x + C$

Where $C$ is the constant of integration [Per the fundamental theorem of calculus regarding indefinite integrals].

Step 4: Algebraic Simplification

Canceling the coefficient $2$ in the first term:

$I = x^2 - 3\sin x + e^x + C$

Final Solution: $x^2 - 3\sin x + e^x + C$

Step 1: Expanding the Integrand

The given integral is: $I = \int \sqrt{x} (3x^2 + 2x + 3) \, dx$

We express $\sqrt{x}$ as a power of $x$, specifically $x^{1/2}$. Applying the Distributive Property of Multiplication over Addition [Per the Distributive Law of Arithmetic]:

$I = \int x^{1/2} (3x^2 + 2x + 3) \, dx$

$I = \int (3x^{2 + 1/2} + 2x^{1 + 1/2} + 3x^{1/2}) \, dx$

$I = \int (3x^{5/2} + 2x^{3/2} + 3x^{1/2}) \, dx$

Step 2: Applying the Linearity Property of Integrals

According to the Linearity of the Integral operator, the integral of a sum is the sum of the integrals [Per the Sum Rule for Integration]:

$I = 3 \int x^{5/2} \, dx + 2 \int x^{3/2} \, dx + 3 \int x^{1/2} \, dx$

Step 3: Calculating Antiderivatives

We apply the Power Rule for integration, which states $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$, provided $n \neq -1$ [Per the Power Rule of Calculus].

• For $\int x^{5/2} \, dx$: $\frac{x^{5/2 + 1}}{5/2 + 1} = \frac{x^{7/2}}{7/2} = \frac{2}{7}x^{7/2}$
• For $\int x^{3/2} \, dx$: $\frac{x^{3/2 + 1}}{3/2 + 1} = \frac{x^{5/2}}{5/2} = \frac{2}{5}x^{5/2}$
• For $\int x^{1/2} \, dx$: $\frac{x^{1/2 + 1}}{1/2 + 1} = \frac{x^{3/2}}{3/2} = \frac{2}{3}x^{3/2}$

Step 4: Final Assembly and Simplification

Substituting the results back into the expression:

$I = 3 \left( \frac{2}{7} x^{7/2} \right) + 2 \left( \frac{2}{5} x^{5/2} \right) + 3 \left( \frac{2}{3} x^{3/2} \right) + C$

$I = \frac{6}{7} x^{7/2} + \frac{4}{5} x^{5/2} + 2 x^{3/2} + C$

Final Solution: $\frac{6}{7} x^{7/2} + \frac{4}{5} x^{5/2} + 2 x^{3/2} + C$

Step 1: Expanding the Integrand

The given integral is $I = \int (\sqrt{x} - \frac{1}{\sqrt{x}})^2 dx$. To evaluate this, we apply the algebraic identity $(a - b)^2 = a^2 - 2ab + b^2$, where $a = \sqrt{x}$ and $b = \frac{1}{\sqrt{x}}$.

Expanding the expression:

Step 2: Linearity Property of Integration

By the linearity property of integrals, $\int [f(x) + g(x) + h(x)] dx = \int f(x) dx + \int g(x) dx + \int h(x) dx$. We distribute the integral across the terms:

Step 3: Applying Standard Integration Formulas

We apply the power rule for integration $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ (for $n \neq -1$) and the logarithmic integration formula $\int \frac{1}{x} dx = \ln|x| + C$.

• For $\int x^1 dx$: $\frac{x^{1+1}}{1+1} = \frac{x^2}{2}$ [Power rule]
• For $\int 2 dx$: $2x$ [Constant rule]
• For $\int \frac{1}{x} dx$: $\ln|x|$ [Logarithmic rule]

Step 4: Compiling the Primitive

Combining the results and adding the constant of integration $C$:

Final Solution: \(\frac{x^2}{2} - 2x + \ln|x| + C\)

Step 1: Theoretical Foundation

To find the anti-derivative of the function $f(x) = (ax + b)^2$ by the method of inspection, we identify a function $F(x)$ such that its derivative $F'(x) = f(x)$. [Per the Fundamental Theorem of Calculus and the definition of an anti-derivative].

Step 2: Identifying the Primitive Function

Recall the Power Rule for differentiation: $\frac{d}{dx} [x^n] = n x^{n-1}$. For a composite function of the form $(ax + b)^n$, we apply the Chain Rule:
$\frac{d}{dx} [(ax + b)^n] = n(ax + b)^{n-1} \cdot \frac{d}{dx}(ax + b) = n(ax + b)^{n-1} \cdot a$

Step 3: Algebraic Manipulation

We require the derivative of our chosen function to be exactly $(ax + b)^2$. Thus, we set $n-1 = 2$, which implies $n = 3$.
Let $F(x) = k(ax + b)^3$, where $k$ is a constant to be determined.
Taking the derivative:
$F'(x) = \frac{d}{dx} [k(ax + b)^3]$
$F'(x) = k \cdot 3(ax + b)^2 \cdot a$
$F'(x) = 3ka(ax + b)^2$

Step 4: Solving for the Constant

We equate the derived expression to our target function:
$3ka(ax + b)^2 = (ax + b)^2$
Dividing both sides by $(ax + b)^2$ (where $ax + b \neq 0$):
$3ka = 1 \implies k = \frac{1}{3a}$

Step 5: Visualizing the Function Growth

Step 6: Final Integration Statement

Substituting $k$ back into the original assumption:
$F(x) = \frac{1}{3a}(ax + b)^3$
Since the derivative of a constant is zero, the general anti-derivative includes an arbitrary constant $C$ [By the definition of the Indefinite Integral].

Final Solution: $\int (ax + b)^2 \, dx = \frac{(ax + b)^3}{3a} + C$

Given Variables & Initial Setup

We are required to find the anti-derivative (integral) of the function $f(x) = \sin 2x - 4e^{3x}$ using the method of inspection. By the method of inspection, we look for a function $F(x)$ such that its derivative $F'(x) = f(x)$.

Step 1: Identifying the Anti-derivative of $\sin 2x$

We know that the derivative of the cosine function relates to the sine function. Specifically, consider the function $\cos 2x$.

Using the chain rule [Per the Chain Rule of Differentiation]:

$\frac{d}{dx}(\cos 2x) = -\sin 2x \cdot \frac{d}{dx}(2x) = -2 \sin 2x$

To isolate $\sin 2x$, we rearrange the equation:

$- \frac{1}{2} \frac{d}{dx}(\cos 2x) = \sin 2x$

$\frac{d}{dx} \left( -\frac{1}{2} \cos 2x \right) = \sin 2x$

Thus, the anti-derivative of $\sin 2x$ is $-\frac{1}{2} \cos 2x$.

Step 2: Identifying the Anti-derivative of $-4e^{3x}$

We know that the derivative of the exponential function $e^{kx}$ is $k e^{kx}$ [Per the Rule of Differentiation for Exponential Functions].

Consider the function $e^{3x}$:

$\frac{d}{dx}(e^{3x}) = 3e^{3x}$

We need to obtain $-4e^{3x}$. We manipulate the expression as follows:

$\frac{1}{3} \frac{d}{dx}(e^{3x}) = e^{3x}$

Multiply both sides by $-4$:

$-4 \left( \frac{1}{3} \frac{d}{dx}(e^{3x}) \right) = -4e^{3x}$

$\frac{d}{dx} \left( -\frac{4}{3} e^{3x} \right) = -4e^{3x}$

Thus, the anti-derivative of $-4e^{3x}$ is $-\frac{4}{3} e^{3x}$.

Step 3: Combining the Results

By the linearity property of differentiation [Per the Linearity of the Derivative Operator], the derivative of the sum is the sum of the derivatives:

$\frac{d}{dx} \left( -\frac{1}{2} \cos 2x - \frac{4}{3} e^{3x} \right) = \frac{d}{dx} \left( -\frac{1}{2} \cos 2x \right) + \frac{d}{dx} \left( -\frac{4}{3} e^{3x} \right)$

Substituting the results from Steps 1 and 2:

$\frac{d}{dx} \left( -\frac{1}{2} \cos 2x - \frac{4}{3} e^{3x} \right) = \sin 2x - 4e^{3x}$

Summary of Calculation

Final Solution: The anti-derivative of $\sin 2x - 4e^{3x}$ is $-\frac{1}{2} \cos 2x - \frac{4}{3} e^{3x} + C$, where $C$ is an arbitrary constant.

Step 1: Application of Linearity Property of Integration

To evaluate the integral $I = \int (2x^2 - 3\sin x + 5\sqrt{x}) dx$, we utilize the linearity property of the indefinite integral, which states that the integral of a sum/difference of functions is the sum/difference of their individual integrals: $$\int [af(x) + bg(x) + ch(x)] dx = a \int f(x) dx + b \int g(x) dx + c \int h(x) dx$$ [By Linearity of Integral Operators]

Thus, we distribute the integral operator across the terms:

$I = 2 \int x^2 dx - 3 \int \sin x dx + 5 \int \sqrt{x} dx$

Step 2: Integration of Individual Terms

We apply the standard power rule of integration, $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ (for $n \neq -1$), and the standard trigonometric integral formula $\int \sin x dx = -\cos x + C$.

• Term 1: $\int x^2 dx = \frac{x^{2+1}}{2+1} = \frac{x^3}{3}$
• Term 2: $\int \sin x dx = -\cos x$
• Term 3: $\int \sqrt{x} dx = \int x^{1/2} dx = \frac{x^{1/2+1}}{1/2+1} = \frac{x^{3/2}}{3/2} = \frac{2}{3} x^{3/2}$

Step 3: Algebraic Assembly and Constant of Integration

Substituting the results from Step 2 into the expression derived in Step 1:

$I = 2 \left( \frac{x^3}{3} \right) - 3 \left( -\cos x \right) + 5 \left( \frac{2}{3} x^{3/2} \right) + C$

Simplifying the coefficients:

$I = \frac{2}{3}x^3 + 3\cos x + \frac{10}{3}x^{3/2} + C$

Visual Representation of the Power Function Component

Final Solution: $\frac{2}{3}x^3 + 3\cos x + \frac{10}{3}x^{3/2} + C$

Step 1: Algebraic Simplification of the Integrand

Given the integral expression: $I = \int \frac{x^3 + 3x + 4}{\sqrt{x}} \, dx$

We express the denominator $\sqrt{x}$ as a power of $x$, specifically $x^{1/2}$. By the laws of exponents, we distribute the denominator across each term in the numerator [Per the Distributive Property of Division]:

$I = \int \frac{x^3}{x^{1/2}} + \frac{3x}{x^{1/2}} + \frac{4}{x^{1/2}} \, dx$

Step 2: Applying Laws of Exponents

We simplify each term using the rule $\frac{x^a}{x^b} = x^{a-b}$:

• Term 1: $\frac{x^3}{x^{1/2}} = x^{3 - 1/2} = x^{5/2}$
• Term 2: $\frac{3x^1}{x^{1/2}} = 3x^{1 - 1/2} = 3x^{1/2}$
• Term 3: $\frac{4}{x^{1/2}} = 4x^{-1/2}$

Thus, the integral becomes:

$I = \int (x^{5/2} + 3x^{1/2} + 4x^{-1/2}) \, dx$

Step 3: Integration using the Power Rule

We apply the linearity property of integrals [Per the Sum Rule of Integration] and the Power Rule, $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ (for $n \neq -1$):

$I = \int x^{5/2} \, dx + 3 \int x^{1/2} \, dx + 4 \int x^{-1/2} \, dx$

Calculating each part:

1. $\int x^{5/2} \, dx = \frac{x^{5/2 + 1}}{5/2 + 1} = \frac{x^{7/2}}{7/2} = \frac{2}{7}x^{7/2}$
2. $3 \int x^{1/2} \, dx = 3 \left( \frac{x^{1/2 + 1}}{1/2 + 1} \right) = 3 \left( \frac{x^{3/2}}{3/2} \right) = 3 \cdot \frac{2}{3} x^{3/2} = 2x^{3/2}$
3. $4 \int x^{-1/2} \, dx = 4 \left( \frac{x^{-1/2 + 1}}{-1/2 + 1} \right) = 4 \left( \frac{x^{1/2}}{1/2} \right) = 4 \cdot 2 x^{1/2} = 8x^{1/2}$

Step 4: Final Compilation and Constant of Integration

Summing the results and appending the arbitrary constant $C$ [Per the Fundamental Theorem of Calculus]:

$I = \frac{2}{7}x^{7/2} + 2x^{3/2} + 8x^{1/2} + C$

Final Solution: $\frac{2}{7}x^{7/2} + 2x^{3/2} + 8x^{1/2} + C$

Step 1: Simplify the Integrand via Polynomial Factoring

Given the integral: $I = \int \frac{x^3 - x^2 + x - 1}{x - 1} dx$

Observe the numerator $P(x) = x^3 - x^2 + x - 1$. We can factor this by grouping terms:

$x^3 - x^2 + x - 1 = x^2(x - 1) + 1(x - 1)$ [Factorization by grouping]

$= (x^2 + 1)(x - 1)$

Step 2: Simplify the Rational Expression

Substitute the factored form back into the integral:

$I = \int \frac{(x^2 + 1)(x - 1)}{x - 1} dx$

Provided $x \neq 1$, we cancel the common term $(x - 1)$ in the numerator and denominator [By the property of rational functions where common factors may be cancelled at points of discontinuity]:

$I = \int (x^2 + 1) dx$

Step 3: Apply the Linearity Property of Integrals

Using the Sum Rule for integration $\int [f(x) + g(x)] dx = \int f(x) dx + \int g(x) dx$:

$I = \int x^2 dx + \int 1 dx$

Step 4: Integrate Using the Power Rule

Apply the Power Rule $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ for $n \neq -1$:

For $\int x^2 dx$: $n = 2$, so $\frac{x^{2+1}}{2+1} = \frac{x^3}{3}$

For $\int 1 dx$: $\int x^0 dx = \frac{x^{0+1}}{0+1} = x$

$I = \frac{x^3}{3} + x + C$

Visualization of the Function

Final Solution: $\frac{x^3}{3} + x + C$

Step 1: Establishing the Theoretical Basis

The method of inspection, also known as the anti-derivative by observation, requires identifying a function $F(x)$ such that its derivative $F'(x)$ equals the given function $f(x)$. In this context, we seek a function whose derivative results in $\sin 2x$. [Fundamental Theorem of Calculus].

Step 2: Selecting the Primary Function

We know from trigonometric differentiation that the derivative of a cosine function yields a negative sine function. Specifically:

$\frac{d}{dx}(\cos 2x) = -2 \sin 2x$ [Chain Rule: $\frac{d}{dx}f(g(x)) = f'(g(x)) \cdot g'(x)$]

Step 3: Algebraic Adjustment for the Target Function

Our objective is to isolate $\sin 2x$. We rearrange the previous derivative equation:

$-2 \sin 2x = \frac{d}{dx}(\cos 2x)$

Dividing both sides by $-2$:

$\sin 2x = -\frac{1}{2} \frac{d}{dx}(\cos 2x)$

Applying the linearity property of derivatives, $\frac{d}{dx}(kf(x)) = k\frac{d}{dx}f(x)$:

$\sin 2x = \frac{d}{dx}\left(-\frac{1}{2} \cos 2x\right)$

Step 4: Vector and Angular Visualization

Step 5: Conclusion via Inspection

Since the derivative of $-\frac{1}{2} \cos 2x$ is $\sin 2x$, by the definition of an anti-derivative (integral), the function $F(x) = -\frac{1}{2} \cos 2x$ satisfies the requirement.

Final Solution: The anti-derivative of $\sin 2x$ is $-\frac{1}{2} \cos 2x$.

Step 1: Establishing the Relationship between Derivative and Integral

Given the derivative of a function $f(x)$ is $\frac{d}{dx} f(x) = 4x^3 - \frac{3}{x^4}$. By the Fundamental Theorem of Calculus, the function $f(x)$ can be recovered by integrating its derivative with respect to $x$:

$f(x) = \int \left( 4x^3 - 3x^{-4} \right) dx$ [Definition of Antiderivative]

Step 2: Performing the Integration

Applying the power rule for integration, $\int x^n dx = \frac{x^{n+1}}{n+1} + C$, where $n \neq -1$:

$f(x) = 4 \left( \frac{x^{3+1}}{3+1} \right) - 3 \left( \frac{x^{-4+1}}{-4+1} \right) + C$

$f(x) = 4 \left( \frac{x^4}{4} \right) - 3 \left( \frac{x^{-3}}{-3} \right) + C$

$f(x) = x^4 + \frac{1}{x^3} + C$ [Algebraic simplification]

Step 3: Determining the Constant of Integration ($C$)

We are given the initial condition $f(2) = 0$. We substitute $x = 2$ into our expression for $f(x)$ to solve for $C$:

$0 = (2)^4 + \frac{1}{(2)^3} + C$

$0 = 16 + \frac{1}{8} + C$

$0 = \frac{128}{8} + \frac{1}{8} + C$

$0 = \frac{129}{8} + C$

$C = -\frac{129}{8}$

Step 4: Formulating the Final Function

Substituting the value of $C$ back into the equation derived in Step 2:

$f(x) = x^4 + \frac{1}{x^3} - \frac{129}{8}$

Final Solution: $f(x) = x^4 + \frac{1}{x^3} - \frac{129}{8}$

Step 1: Decomposition of the Rational Function

Given the integral: $I = \int \frac{x^3 + 5x^2 - 4}{x^2} dx$

To evaluate this, we distribute the denominator $x^2$ to each term in the numerator. This is permissible as $x \neq 0$ [By the property of algebraic division of polynomials]:

$I = \int \left( \frac{x^3}{x^2} + \frac{5x^2}{x^2} - \frac{4}{x^2} \right) dx$

Step 2: Simplification of Terms

Apply the laws of exponents $\frac{x^a}{x^b} = x^{a-b}$ and $\frac{1}{x^n} = x^{-n}$ to simplify the integrand:

$I = \int \left( x + 5 - 4x^{-2} \right) dx$

Step 3: Applying Linearity of Integration

The integral of a sum/difference is the sum/difference of the integrals [Per the Sum/Difference Rule of Integration]. We split the integral into individual components:

$I = \int x \, dx + \int 5 \, dx - \int 4x^{-2} \, dx$

Step 4: Integration Using Power Rule

Apply the Power Rule $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ for $n \neq -1$:

• $\int x^1 dx = \frac{x^{1+1}}{1+1} = \frac{x^2}{2}$
• $\int 5 dx = 5x$
• $\int 4x^{-2} dx = 4 \left( \frac{x^{-2+1}}{-2+1} \right) = 4 \left( \frac{x^{-1}}{-1} \right) = -4x^{-1}$

Step 5: Final Assembly and Integration Constant

Combine the results and include the constant of integration $C$:

$I = \frac{x^2}{2} + 5x - (-4x^{-1}) + C$

$I = \frac{x^2}{2} + 5x + \frac{4}{x} + C$

Final Solution: $\frac{x^2}{2} + 5x + \frac{4}{x} + C$

Step 1: Algebraic Simplification of the Integrand

Given the integral: $I = \int x^2\left(1 - \frac{1}{x^2}\right) dx$

To evaluate this integral, we first distribute the term $x^2$ into the binomial expression within the parentheses [By Distributive Property of Multiplication over Subtraction].

$I = \int \left( x^2 \cdot 1 - x^2 \cdot \frac{1}{x^2} \right) dx$

$I = \int \left( x^2 - 1 \right) dx$

Step 2: Application of Linearity of Integration

The integral of a sum or difference of functions is equal to the sum or difference of their individual integrals [Per the Linearity Property of Integration].

$I = \int x^2 dx - \int 1 dx$

Step 3: Execution of Basic Integration Rules

Apply the Power Rule for integration, which states $\int x^n dx = \frac{x^{n+1}}{n+1} + C$, where $n \neq -1$.

• For $\int x^2 dx$: Here $n=2$, thus $\frac{x^{2+1}}{2+1} = \frac{x^3}{3}$.
• For $\int 1 dx$: Since $1 = x^0$, we have $\frac{x^{0+1}}{0+1} = x$.

Combining these results:

$I = \frac{x^3}{3} - x + C$

Where $C$ represents the constant of integration required for indefinite integrals [Fundamental Theorem of Calculus].

Visual Representation of the Primitive Function

Final Solution: $\frac{x^3}{3} - x + C$

Step 1: Distribution of the Integrand

Given the indefinite integral: $I = \int \sec x (\sec x + \tan x) \, dx$

Distribute $\sec x$ across the terms within the parentheses [Per the Distributive Property of Multiplication]:

Step 2: Applying the Linearity of Integration

By the additive property of indefinite integrals, the integral of a sum is equal to the sum of the integrals [Per the Linearity Property]:

Step 3: Evaluating Fundamental Integrals

Utilizing standard trigonometric integration formulas:

• The antiderivative of $\sec^2 x$ is $\tan x$, because $\frac{d}{dx}(\tan x) = \sec^2 x$.
• The antiderivative of $\sec x \tan x$ is $\sec x$, because $\frac{d}{dx}(\sec x) = \sec x \tan x$.

Substituting these into the expression:

Where $C$ represents the constant of integration [Per the Fundamental Theorem of Calculus].

Visual Representation of Integral Components

Final Solution: $\tan x + \sec x + C$

Step 1: Application of Linearity Properties of Integration

To evaluate the integral $\int (ax^2 + bx + c) \, dx$, we utilize the Linearity Property of Integration. This property states that the integral of a sum of functions is the sum of the integrals of individual functions, and constant multipliers can be extracted outside the integral sign [Per the Linearity Property: $\int [k_1 f(x) + k_2 g(x) + k_3 h(x)] \, dx = k_1 \int f(x) \, dx + k_2 \int g(x) \, dx + k_3 \int h(x) \, dx$].

Expanding the expression:

$\int (ax^2 + bx + c) \, dx = a \int x^2 \, dx + b \int x \, dx + c \int 1 \, dx$

Step 2: Applying the Power Rule for Integration

We apply the Power Rule for Integration, which is defined as $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$, where $n \neq -1$ [By the Power Rule for Antiderivatives].

• For the term $\int x^2 \, dx$: $n=2$, thus $\int x^2 \, dx = \frac{x^{2+1}}{2+1} = \frac{x^3}{3}$.
• For the term $\int x^1 \, dx$: $n=1$, thus $\int x^1 \, dx = \frac{x^{1+1}}{1+1} = \frac{x^2}{2}$.
• For the term $\int 1 \, dx$: This is $\int x^0 \, dx$, thus $\int x^0 \, dx = \frac{x^{0+1}}{0+1} = x$.

Step 3: Synthesis and Final Formulation

Substituting the results from Step 2 back into the linear expression obtained in Step 1:

Result $= a \left( \frac{x^3}{3} \right) + b \left( \frac{x^2}{2} \right) + c(x) + C$

Where $C$ is the constant of integration, representing the family of antiderivatives [By the Fundamental Theorem of Calculus].

Final Solution: $\frac{ax^3}{3} + \frac{bx^2}{2} + cx + C$

Step 1: Application of Linearity Property of Integrals

Given the integral: $\int (4e^{3x} + 1) \, dx$.

By the linearity property of integration [Ref: $\int [af(x) + bg(x)] \, dx = a\int f(x) \, dx + b\int g(x) \, dx$], we can distribute the integral operator across the sum:

$\int (4e^{3x} + 1) \, dx = \int 4e^{3x} \, dx + \int 1 \, dx$

$= 4 \int e^{3x} \, dx + \int 1 \, dx$

Step 2: Execution of Elementary Integration

We apply the standard integral formulas:

• For the exponential term: $\int e^{ax} \, dx = \frac{1}{a}e^{ax} + C$ [Ref: Integration rule for exponential functions where $a \neq 0$]
• For the constant term: $\int 1 \, dx = x + C$ [Ref: Integration rule for power functions where $x^n$ for $n=0$]

Applying these to our expression:

$4 \left( \frac{e^{3x}}{3} \right) + x + C$

Step 3: Algebraic Simplification

Combine the terms to arrive at the final antiderivative:

$\frac{4}{3}e^{3x} + x + C$

Where $C$ represents the constant of integration, accounting for the family of curves that satisfy the derivative condition [Ref: Fundamental Theorem of Calculus].

Final Solution: $\frac{4}{3}e^{3x} + x + C$

Step 1: Trigonometric Simplification

To evaluate the integral $\int \frac{\sec^2 x}{\csc^2 x} dx$, we must first express the integrand in terms of fundamental trigonometric functions, namely $\sin x$ and $\cos x$.

Using the reciprocal identities:

• $\sec x = \frac{1}{\cos x} \implies \sec^2 x = \frac{1}{\cos^2 x}$ [Reciprocal Identity]
• $\csc x = \frac{1}{\sin x} \implies \csc^2 x = \frac{1}{\sin^2 x}$ [Reciprocal Identity]

Substituting these into the integrand:

$\frac{\sec^2 x}{\csc^2 x} = \frac{\frac{1}{\cos^2 x}}{\frac{1}{\sin^2 x}} = \frac{1}{\cos^2 x} \cdot \frac{\sin^2 x}{1} = \frac{\sin^2 x}{\cos^2 x}$

Applying the quotient identity, $\frac{\sin x}{\cos x} = \tan x$, we obtain:

$\frac{\sin^2 x}{\cos^2 x} = \tan^2 x$ [Quotient Identity]

Step 2: Algebraic Manipulation for Integration

The standard integral table does not provide a direct primitive for $\tan^2 x$. We utilize the Pythagorean identity to express $\tan^2 x$ in a form that is directly integrable:

$1 + \tan^2 x = \sec^2 x \implies \tan^2 x = \sec^2 x - 1$ [Pythagorean Identity]

Thus, the integral becomes:

$I = \int (\sec^2 x - 1) dx$

Step 3: Execution of the Integration

By the Linearity Property of integrals, we split the expression into two separate integrals:

$I = \int \sec^2 x \, dx - \int 1 \, dx$

Applying the fundamental integration rules:

• $\int \sec^2 x \, dx = \tan x + C_1$ [Standard Integral Formula]
• $\int 1 \, dx = x + C_2$ [Power Rule: $\int x^n dx = \frac{x^{n+1}}{n+1}$ where $n=0$]

Combining these, where $C = C_1 - C_2$ is the constant of integration:

$I = \tan x - x + C$

Visual Representation of the Integrand Transformation

Final Solution: $\tan x - x + C$

Given Variables & Initial Setup

We are tasked with finding the antiderivative (indefinite integral) of the function $f(x) = \sqrt{x} + \frac{1}{\sqrt{x}}$.

The integral to be evaluated is:
$I = \int \left( \sqrt{x} + \frac{1}{\sqrt{x}} \right) dx$

Step 1: Algebraic Simplification

To integrate, we express the function using fractional exponents [Per the Laws of Exponents: $\sqrt[n]{x^m} = x^{m/n}$ and $\frac{1}{x^n} = x^{-n}$]:

$\sqrt{x} = x^{1/2}$

$\frac{1}{\sqrt{x}} = \frac{1}{x^{1/2}} = x^{-1/2}$

Substituting these into the integral:
$I = \int (x^{1/2} + x^{-1/2}) dx$

Step 2: Application of Linearity and Power Rule

We utilize the Sum Rule of integration: $\int [f(x) + g(x)] dx = \int f(x) dx + \int g(x) dx$.

$I = \int x^{1/2} dx + \int x^{-1/2} dx$

Applying the Power Rule of integration, which states $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ for $n \neq -1$:

For the first term ($n = 1/2$):
$\int x^{1/2} dx = \frac{x^{1/2 + 1}}{1/2 + 1} = \frac{x^{3/2}}{3/2} = \frac{2}{3}x^{3/2}$

For the second term ($n = -1/2$):
$\int x^{-1/2} dx = \frac{x^{-1/2 + 1}}{-1/2 + 1} = \frac{x^{1/2}}{1/2} = 2x^{1/2}$

Step 3: Assembly and Constant Integration

Combining the results and adding the constant of integration $C$:

$I = \frac{2}{3}x^{3/2} + 2x^{1/2} + C$

Visual Representation of the Power Function Integration

Final Solution

Final Solution: $\frac{2}{3}x^{3/2} + 2\sqrt{x} + C$

Step 1: Theoretical Foundation of Anti-differentiation

To find the anti-derivative (also known as the indefinite integral) of a function $f(x)$ by the method of inspection, we must identify a function $F(x)$ such that its derivative yields the original function: $$\frac{d}{dx}[F(x)] = f(x)$$ Given $f(x) = e^{2x}$, we seek $F(x)$ such that $\frac{d}{dx}[F(x)] = e^{2x}$.

Step 2: Analysis of Exponential Derivatives

Recall the general rule for differentiating exponential functions involving a constant coefficient in the exponent. According to the Chain Rule [Per the Chain Rule of Differentiation]: $$\frac{d}{dx}[e^{ax}] = a \cdot e^{ax}$$ Where $a$ is a non-zero constant. In the current problem, $a = 2$.

Step 3: Algebraic Manipulation for Equality

Applying the rule to our specific case: $$\frac{d}{dx}[e^{2x}] = 2 \cdot e^{2x}$$ To isolate the term $e^{2x}$, we divide both sides of the equality by the constant $2$ [By the linearity property of derivatives]: $$\frac{1}{2} \cdot \frac{d}{dx}[e^{2x}] = e^{2x}$$ $$\frac{d}{dx}\left[ \frac{1}{2}e^{2x} \right] = e^{2x}$$

Step 4: Verification of the Anti-derivative

Since the derivative of $\frac{1}{2}e^{2x}$ is $e^{2x}$, the function $\frac{1}{2}e^{2x}$ is an anti-derivative of $e^{2x}$.

Step 5: Final Integral Representation

By the definition of indefinite integrals, adding an arbitrary constant $C$ accounts for all possible anti-derivatives: $$\int e^{2x} \, dx = \frac{1}{2}e^{2x} + C$$

Final Solution: $\frac{1}{2}e^{2x} + C$