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Q6:
Integrate the functions in Exercises 1 to 37:
6. $\sqrt{ax+b}$
Solution :
Step 1: Formal Representation of the Integral
We are required to evaluate the indefinite integral of the function $f(x) = \sqrt{ax+b}$ with respect to $x$. Mathematically, this is expressed as:
$I = \int \sqrt{ax+b} \, dx$
Rewrite the radical expression using fractional exponent notation [Per the Laws of Exponents]:
$I = \int (ax+b)^{1/2} \, dx$
Step 2: Method of Substitution
To simplify the integrand, perform a variable substitution to transform the linear expression into a standard power form.
Let $u = ax + b$
Differentiate both sides with respect to $x$ to find the differential relationship [By the Chain Rule of differentiation]:
$\frac{du}{dx} = \frac{d}{dx}(ax + b)$
$\frac{du}{dx} = a$
$dx = \frac{du}{a}$
Step 3: Evaluating the Transformed Integral
Substitute $u$ and $dx$ into the original integral expression:
$I = \int u^{1/2} \cdot \frac{du}{a}$
$I = \frac{1}{a} \int u^{1/2} \, du$
Apply the Power Rule for Integration, $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$, where $n = 1/2$:
$I = \frac{1}{a} \left( \frac{u^{1/2 + 1}}{1/2 + 1} \right) + C$
$I = \frac{1}{a} \left( \frac{u^{3/2}}{3/2} \right) + C$
$I = \frac{1}{a} \cdot \frac{2}{3} u^{3/2} + C$
$I = \frac{2}{3a} u^{3/2} + C$
Step 4: Back-Substitution
Replace $u$ with the original function $ax + b$ to obtain the final primitive in terms of $x$:
$I = \frac{2}{3a} (ax+b)^{3/2} + C$
Visualization of the Integral Curve
Final Solution: $\frac{2}{3a} (ax+b)^{3/2} + C$, where $C$ is the constant of integration.
More Questions from Class 12 Mathematics Integrals EXERCISE 7.2
- Q1: Integrate the functions in Exercises 1 to 37: 1. $\frac{2x}{1+x^2}$
- Q2: Integrate the functions in Exercises 1 to 37: 2. $\frac{(\log x)^2}{x}$
- Q3: Integrate the functions in Exercises 1 to 37: 3. $\frac{1}{x + x \log x}$
- Q4: Integrate the functions in Exercises 1 to 37: 4. $\sin x \sin (\cos x)$
- Q5: Integrate the functions in Exercises 1 to 37: 5. $\sin (ax + b) \cos (ax + b)$
- Q7: Integrate the functions in Exercises 1 to 37: 7. $x \sqrt{x+2}$
- Q8: Integrate the functions in Exercises 1 to 37: 8. $x \sqrt{1+2x^2}$
- Q9: Integrate the functions in Exercises 1 to 37: 9. $(4x+2)\sqrt{x^2+x+1}$
- Q10: Integrate the functions in Exercises 1 to 37: 10. $\frac{1}{x - \sqrt{x}}$
- Q11: Integrate the functions in Exercises 1 to 37: 11. $\frac{x}{\sqrt{x+4}}$, $x > 0$
- Q12: Integrate the functions in Exercises 1 to 37: 12. $(x^3 - 1)^{\frac{1}{3}} x^5$
- Q13: Integrate the functions in Exercises 1 to 37: 13. $\frac{x^2}{(2+3x^3)^3}$
- Q14: Integrate the functions in Exercises 1 to 37: 14. $\frac{1}{x(\log x)^m}$, $x > 0, m \neq 1$
- Q15: Integrate the functions in Exercises 1 to 37: 15. $\frac{x}{9-4x^2}$
- Q16: Integrate the functions in Exercises 1 to 37: 16. $e^{2x+3}$
- Q17: Integrate the functions in Exercises 1 to 37: 17. $\frac{x}{e^{x^2}}$
- Q18: Integrate the functions in Exercises 1 to 37: 18. $\frac{e^{\tan^{-1} x}}{1+x^2}$
- Q19: Integrate the functions in Exercises 1 to 37: 19. $\frac{e^{2x}-1}{e^{2x}+1}$
- Q20: Integrate the functions in Exercises 1 to 37: 20. $\frac{e^{2x}-e^{-2x}}{e^{2x}+e^{-2x}}$
- Q21: Integrate the functions in Exercises 1 to 37: 21. $\tan^2(2x-3)$
- Q22: Integrate the functions in Exercises 1 to 37: 22. $\sec^2(7-4x)$
- Q23: Integrate the functions in Exercises 1 to 37: 23. $\frac{\sin^{-1} x}{\sqrt{1-x^2}}$
- Q24: Integrate the functions in Exercises 1 to 37: 24. $\frac{2\cos x - 3\sin x}{6\cos x + 4\sin x}$
- Q25: Integrate the functions in Exercises 1 to 37: 25. $\frac{1}{\cos^2 x (1-\tan x)^2}$
- Q26: Integrate the functions in Exercises 1 to 37: 26. $\frac{\cos \sqrt{x}}{\sqrt{x}}$
- Q27: Integrate the functions in Exercises 1 to 37: 27. $\sqrt{\sin 2x} \cos 2x$
- Q28: Integrate the functions in Exercises 1 to 37: 28. $\frac{\cos x}{\sqrt{1+\sin x}}$
- Q29: Integrate the functions in Exercises 1 to 37: 29. $\cot x \log \sin x$
- Q30: Integrate the functions in Exercises 1 to 37: 30. $\frac{\sin x}{1+\cos x}$
- Q31: Integrate the functions in Exercises 1 to 37: 31. $\frac{\sin x}{(1+\cos x)^2}$
- Q32: Integrate the functions in Exercises 1 to 37: 32. $\frac{1}{1+\cot x}$
- Q33: Integrate the functions in Exercises 1 to 37: 33. $\frac{1}{1-\tan x}$
- Q34: Integrate the functions in Exercises 1 to 37: 34. $\frac{\sqrt{\tan x}}{\sin x \cos x}$
- Q35: Integrate the functions in Exercises 1 to 37: 35. $\frac{(1+\log x)^2}{x}$
- Q36: Integrate the functions in Exercises 1 to 37: 36. $\frac{(x+1)(x+\log x)^2}{x}$
- Q37: Integrate the functions in Exercises 1 to 37: 37. $\frac{x^3 \sin(\tan^{-1} x^4)}{1+x^8}$
- Q38: Choose the correct answer in Exercises 38 and 39. 38. $\int \frac{10x^9 + 10^x \log_e 10}{x^{10} + 10^x} dx$ equals
- Q39: Choose the correct answer in Exercises 38 and 39. 39. $\int \frac{dx}{\sin^2 x \cos^2 x}$ equals
CBSE Solutions for Class 12 Mathematics Integrals
- Integrals Class 12 EXERCISE 7.1
- Integrals Class 12 EXERCISE 7.10
- 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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