CBSE Class 12 Mathematics Integrals Assignment Set 07

Practice Questions with Answers

Question 1. Evaluate \( \int \tan^{-1}\sqrt{x}\,dx \).
Answer:
\( (x + 1)\tan^{-1}\sqrt{x} - \sqrt{x} + C \)

Question 2. Evaluate \( \int \frac{x^3}{x + 1}\,dx \).
Answer:
\( x - \frac{x^2}{2} + \frac{x^3}{3} - \log|1 + x| + C \)

Question 3. Evaluate \( \int_{0}^{\pi/2} \cos x\,e^{\sin x}\,dx \).
Answer:
e − 1

Question 4. Evaluate \( \int \frac{x + 3}{(x + 4)^2}e^x\,dx \).
Answer:
\( e^x\left(\frac{1}{x + 4}\right) + C \)

Question 5. Find the value of \( \int_{2}^{5} \frac{\sqrt{x}}{\sqrt{x} + \sqrt{7 - x}}\,dx \).
Answer:
\( \frac{3}{2} \)

Question 6. Evaluate \( \int \frac{dx}{\cos x + \sqrt{3}\sin x} \).
Answer:
\( \frac{1}{2}\log\left|\tan\left(\frac{x}{2} + \frac{\pi}{12}\right)\right| + C \)

Question 7. If \( \int_{0}^{a} \frac{1}{1 + 4x^2}\,dx = \frac{\pi}{8} \), then find the value of a.
Answer:
\( \frac{1}{2} \)

Question 8. Evaluate \( \int \frac{\sin x}{3 + 4\cos^2 x}\,dx \).
Answer:
\( -\frac{1}{2\sqrt{3}}\tan^{-1}\left(\frac{2\cos x}{\sqrt{3}}\right) + C \)

Question 9. Find the value of \( \int_{-\pi}^{\pi} \sin^3 x\cos^2 x\,dx \).
Answer:
0

Question 10. Evaluate \( \int \sin^{-1} x\,dx \).
Answer:
\( x\sin^{-1} x + \sqrt{1 - x^2} + C \)

Question 11. Evaluate \( \int \frac{(x^2 + 2)}{x + 1}\,dx \).
Answer:
\( \frac{x^2}{2} - x + 3\log|x + 1| + C \)

Question 12. Evaluate \( \int \frac{(1 + \cos x)}{x + \sin x}\,dx \).
Answer:
\( \log|x + \sin x| + C \)

Question 13. Evaluate \( \int \frac{\cos x - \sin x}{\sqrt{1 + \sin 2x}}\,dx \).
Answer:
\( \log|\sin x + \cos x| + C \)

Question 14. Evaluate \( \int \frac{dx}{\sqrt{16 - 9x^2}} \).
Answer:
\( \frac{1}{3}\sin^{-1}\left(\frac{3x}{4}\right) + C \)

Question 15. Evaluate \( \int \sqrt{x} \cdot \log x\,dx \).
Answer:
\( \frac{2}{3}x^{3/2}\left(\log x - \frac{2}{3}\right) + C \)