Question

Match the integrals in Column I with the values in Column II and indicate your answer by darkening the appropriate
bubbles in the 4 4 matrix given in the ORS.

Answer 1

Let $A={\int }_{-1}^1\frac{dx}{1+x^2}$
Put $x=\tan \theta \Rightarrow dx={\sec }^2\theta d\theta$
$\therefore A=2{\int }_0^{\frac{\pi }{4}}d\theta =\frac{\pi }{2}$
Let $B={\int }_0^1\frac{dx}{\sqrt{1-x^2}}$
Put $x=\sin \theta \Rightarrow dx=\cos \theta d\theta$
$\therefore B={\int }_0^{\frac{\pi }{2}}d\theta =\frac{\pi }{2}$
$C={\int }_2^3\frac{dx}{1-x^2}=\frac{1}{2}{\left[\log \left(\frac{1+x}{1-x}\right)\right]}_2^3$
$=\frac{1}{2}[\log \left(\frac{4}{-2}\right)-\log \left(\frac{3}{-1}\right)]=\frac{1}{2}\ln \frac{2}{3}$
$D={\int }_1^2\frac{dx}{x\sqrt{x^2-1}}={\left[{\sec }^{-1}x\right]}_1^2=\frac{\pi }{3}-0=\frac{\pi }{3}$