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Q16:
Integrate the functions in Exercises 1 to 23.
16. $\frac{4x+1}{\sqrt{2x^2+x-3}}$
Solution :
Given Integral
Evaluate the integral: $I = \int \frac{4x+1}{\sqrt{2x^2+x-3}} \, dx$
Step 1: Expressing the Numerator as a Derivative of the Quadratic Under the Radical
Let $f(x) = 2x^2 + x - 3$. The derivative of $f(x)$ is $f'(x) = 4x + 1$. [By the substitution rule for differentiation].
We observe that the numerator is exactly the derivative of the expression inside the square root. Therefore, we can express the integral in terms of $f(x)$ and $f'(x)$.
Step 2: Substitution Method
Let $t = 2x^2 + x - 3$.
Differentiating both sides with respect to $x$: $\frac{dt}{dx} = 4x + 1 \implies dt = (4x + 1) \, dx$. [By the Chain Rule of Calculus].
Substituting these into the original integral:
$I = \int \frac{dt}{\sqrt{t}}$
$I = \int t^{-1/2} \, dt$
Step 3: Evaluating the Integral
Apply the power rule for integration: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$. [By the Fundamental Theorem of Calculus].
$I = \frac{t^{-1/2 + 1}}{-1/2 + 1} + C$
$I = \frac{t^{1/2}}{1/2} + C$
$I = 2\sqrt{t} + C$
Step 4: Back-Substitution
Substitute $t = 2x^2 + x - 3$ back into the expression:
$I = 2\sqrt{2x^2 + x - 3} + C$
Visual Representation of the Function
Final Solution: $2\sqrt{2x^2+x-3} + C$
More Questions from Class 12 Mathematics Integrals EXERCISE 7.4
- Q1: Integrate the functions in Exercises 1 to 23. 1. $\frac{3x^2}{x^6+1}$
- Q2: Integrate the functions in Exercises 1 to 23. 2. $\frac{1}{\sqrt{1+4x^2}}$
- Q3: Integrate the functions in Exercises 1 to 23. 3. $\frac{1}{\sqrt{(2-x)^2+1}}$
- Q4: Integrate the functions in Exercises 1 to 23. 4. $\frac{1}{\sqrt{9-25x^2}}$
- Q5: Integrate the functions in Exercises 1 to 23. 5. $\frac{3x}{1+2x^4}$
- Q6: Integrate the functions in Exercises 1 to 23. 6. $\frac{x^2}{1-x^6}$
- Q7: Integrate the functions in Exercises 1 to 23. 7. $\frac{x-1}{\sqrt{x^2-1}}$
- Q8: Integrate the functions in Exercises 1 to 23. 8. $\frac{x^2}{\sqrt{x^6+a^6}}$
- Q9: Integrate the functions in Exercises 1 to 23. 9. $\frac{\sec^2 x}{\sqrt{\tan^2 x + 4}}$
- Q10: Integrate the functions in Exercises 1 to 23. 10. $\frac{1}{\sqrt{x^2+2x+2}}$
- Q11: Integrate the functions in Exercises 1 to 23. 11. $\frac{1}{9x^2+6x+5}$
- Q12: Integrate the functions in Exercises 1 to 23. 12. $\frac{1}{\sqrt{7-6x-x^2}}$
- Q13: Integrate the functions in Exercises 1 to 23. 13. $\frac{1}{\sqrt{(x-1)(x-2)}}$
- Q14: Integrate the functions in Exercises 1 to 23. 14. $\frac{1}{\sqrt{8+3x-x^2}}$
- Q15: Integrate the functions in Exercises 1 to 23. 15. $\frac{1}{\sqrt{(x-a)(x-b)}}$
- Q17: Integrate the functions in Exercises 1 to 23. 17. $\frac{x+2}{\sqrt{x^2-1}}$
- Q18: Integrate the functions in Exercises 1 to 23. 18. $\frac{5x-2}{1+2x+3x^2}$
- Q19: Integrate the functions in Exercises 1 to 23. 19. $\frac{6x+7}{\sqrt{(x-5)(x-4)}}$
- Q20: Integrate the functions in Exercises 1 to 23. 20. $\frac{x+2}{\sqrt{4x-x^2}}$
- Q21: Integrate the functions in Exercises 1 to 23. 21. $\frac{x+2}{\sqrt{x^2+2x+3}}$
- Q22: Integrate the functions in Exercises 1 to 23. 22. $\frac{x+3}{x^2-2x-5}$
- Q23: Integrate the functions in Exercises 1 to 23. 23. $\frac{5x+3}{\sqrt{x^2+4x+10}}$
- Q24: Choose the correct answer in Exercises 24 and 25. 24. $\int \frac{dx}{x^2+2x+2}$ equals
- Q25: Choose the correct answer in Exercises 24 and 25. 25. $\int \frac{dx}{\sqrt{9x-4x^2}}$ equals
CBSE Solutions for Class 12 Mathematics Integrals
- Integrals Class 12 EXERCISE 7.1
- Integrals Class 12 EXERCISE 7.10
- Integrals Class 12 EXERCISE 7.2
- Integrals Class 12 EXERCISE 7.3
- 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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