NCERT Solutions
Class 12 - Mathematics Part-2 - Chapter 7: INTEGRALS
Exercise 7.9

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Question. 1

Evaluate the integral \(\int_{0}^{1} \frac{x}{x^2+1}\,dx\).

Answer:

\(\frac{1}{2}\log 2\)

Question. 2

Evaluate the integral \(\int_{0}^{\pi/2} \sqrt{\sin\phi\,\cos^5\phi}\,d\phi\).

Answer:

\(\frac{64}{231}\)

Question. 3

Evaluate the integral \(\int_{0}^{1} \sin^{-1}\!\left(\frac{2x}{1+x^2}\right)\,dx\).

Answer:

\(\frac{\pi}{2}-\log 2\)

Question. 4

Evaluate the integral \(\int_{0}^{2} x\sqrt{x+2}\,dx\) (Put \(x+2=t^2\)).

Answer:

\(\frac{16\sqrt{2}}{15}(\sqrt{2}+1)\)

Question. 5

Evaluate the integral \(\int_{0}^{\pi/2} \frac{\sin x}{1+\cos^2 x}\,dx\).

Answer:

\(\frac{\pi}{4}\)

Question. 6

Evaluate the integral \(\int_{0}^{2} \frac{dx}{x+4-x^2}\).

Answer:

\(\frac{1}{\sqrt{17}}\log\left(\frac{21+5\sqrt{17}}{4}\right)\)

Question. 7

Evaluate the integral \(\int_{-1}^{1} \frac{dx}{x^2+2x+5}\).

Answer:

\(\frac{\pi}{8}\)

Question. 8

Evaluate the integral \(\int_{1}^{2} \left(\frac{1}{x}-\frac{1}{2x^2}\right)e^{2x}\,dx\).

Answer:

\(\frac{e^2(e^2-2)}{4}\)

Question.  9

The value of the integral \(\int_{1/3}^{1} \frac{(x-x^3)^{1/3}}{x^4}\,dx\) is

(a)

\(6\)

(b)

\(0\)

(c)

\(3\)

(d)

\(4\)

Question.  10

If \(f(x)=\int_{0}^{x} t\sin t\,dt\), then \(f'(x)\) is

(a)

\(\cos x + x\sin x\)

(b)

\(x\sin x\)

(c)

\(x\cos x\)

(d)

\(\sin x + x\cos x\)