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Q5:
Find an anti derivative (or integral) of the following functions by the method of inspection.
5. $\sin 2x - 4e^{3x}$
Solution :
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
| Function Component | Inspection Function | Resulting Anti-derivative |
|---|---|---|
| $\sin 2x$ | $\cos 2x$ | $-\frac{1}{2} \cos 2x$ |
| $-4e^{3x}$ | $e^{3x}$ | $-\frac{4}{3} e^{3x}$ |
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.
More Questions from Class 12 Mathematics Integrals EXERCISE 7.1
- Q1: Find an anti derivative (or integral) of the following functions by the method of inspection. 1. $\sin 2x$
- Q2: Find an anti derivative (or integral) of the following functions by the method of inspection. 2. $\cos 3x$
- Q3: Find an anti derivative (or integral) of the following functions by the method of inspection. 3. $e^{2x}$
- Q4: Find an anti derivative (or integral) of the following functions by the method of inspection. 4. $(ax + b)^2$
- Q6: Find the following integrals in Exercises 6 to 20: 6. $\int (4e^{3x} + 1) dx$
- Q7: Find the following integrals in Exercises 6 to 20: 7. $\int x^2(1 - \frac{1}{x^2}) dx$
- Q8: Find the following integrals 8. $\int (ax^2 + bx + c) dx$
- Q9: Find the following integrals 9. $\int (2x^2 + e^x) dx$
- Q10: Find the following integrals 10. $\int (\sqrt{x} - \frac{1}{\sqrt{x}})^2 dx$
- Q11: Find the following integrals 11. $\int \frac{x^3 + 5x^2 - 4}{x^2} dx$
- Q12: Find the following integrals 12. $\int \frac{x^3 + 3x + 4}{\sqrt{x}} dx$
- Q13: Find the following integrals 13. $\int \frac{x^3 - x^2 + x - 1}{x - 1} dx$
- Q14: Find the following integrals 14. $\int (1 - x)\sqrt{x} dx$
- Q15: Find the following integrals 15. $\int \sqrt{x} (3x^2 + 2x + 3) dx$
- Q16: Find the following integrals 16. $\int (2x - 3\cos x + e^x) dx$
- Q17: Find the following integrals 17. $\int (2x^2 - 3\sin x + 5\sqrt{x}) dx$
- Q18: Find the following integrals 18. $\int \sec x (\sec x + \tan x) dx$
- Q19: Find the following integrals 19. $\int \frac{\sec^2 x}{\csc^2 x} dx$
- Q20: Find the following integrals 20. $\int \frac{2 - 3\sin x}{\cos^2 x} dx$
- Q21: Choose the correct answer 21. The anti derivative of $(\sqrt{x} + \frac{1}{\sqrt{x}})$ equals
- Q22: Choose the correct answer 22. If $\frac{d}{dx} f(x) = 4x^3 - \frac{3}{x^4}$ such that $f(2) = 0$. Then $f(x)$ is
CBSE Solutions for Class 12 Mathematics Integrals
- Integrals Class 12 EXERCISE 7.10
- Integrals Class 12 EXERCISE 7.2
- Integrals Class 12 EXERCISE 7.3
- Integrals Class 12 EXERCISE 7.4
- Integrals Class 12 EXERCISE 7.5
- Integrals Class 12 EXERCISE 7.6
- Integrals Class 12 EXERCISE 7.7
- Integrals Class 12 EXERCISE 7.8
- Integrals Class 12 EXERCISE 7.9
- Integrals Class 12 Miscellaneous Exercise on Chapter 7
Chapters in CBSE - Class 12 Mathematics
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