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Q4:
Find an anti derivative (or integral) of the following functions by the method of inspection.
4. $(ax + b)^2$
Solution :
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: $\int (ax + b)^2 \, dx = \frac{(ax + b)^3}{3a} + C$
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. $e^{2x}$
- Q5: Find an anti derivative (or integral) of the following functions by the method of inspection. 5. $\sin 2x - 4e^{3x}$
- 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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