If then match the entries of Column I with Column II considering as an arbitrary…

MATHEMATICS · Subjective · A Level · Mathematics · Integration · Integration

Problem

x \in (0, 1)

then match the entries of Column I with Column II considering

C

as an arbitrary constant of integration.

Column I	Column II
(A) $\int \tan \left( 2 \tan^{-1} \frac{\sqrt{1+\sqrt{x}} - 1}{\sqrt{1+\sqrt{x}} + 1} \right) dx$	(p) $\frac{4}{3} x^{3/4} + C$
(B) $\int \cot \left( 2 \tan^{-1} \frac{\sqrt{1+\sqrt{x}} - \sqrt[4]{x}}{\sqrt{1+\sqrt{x}} + \sqrt[4]{x}} \right) dx$	(q) $\frac{4}{5} x^{5/4} + C$
(C) $\int \frac{1 - \tan \left( \frac{1}{2} \sin^{-1} \left( \frac{1 - \sqrt{x}}{1 + \sqrt{x}} \right) \right)}{1 + \tan \left( \frac{1}{2} \sin^{-1} \left( \frac{1 - \sqrt{x}}{1 + \sqrt{x}} \right) \right)} dx$	(r) $\frac{2}{3} x^{3/4} + C$
(D) $\int \sqrt{x} \tan \left( 2 \tan^{-1} \left( \frac{\sqrt{1+\sqrt{x}} + 1 - \sqrt{1+\sqrt{x}} - 1}{\sqrt{1+\sqrt{x}} + 1 + \sqrt{1+\sqrt{x}} - 1} \right) \right) dx$	(s) $\frac{2}{5} x^{5/4} + C$