Question

$\int x\sec x^{2}dx\mathrm{is}\mathrm{equal}\mathrm{to}$

(a) $\frac{1}{2}$ log (sec x² + tan x²) + C

(b) $\frac{x^2}{2}$ log (sec x² + tan x²) + C

(c) 2 log (sec x² + tan x²) + C

(d) none of these

Answer 1

(a) $\frac{1}{2}$ log (sec x² + tan x²) + C

$$\begin{gathered} \mathrm{Let}I=\int x\sec x^{2}dx \\ \mathrm{Putting}{x}^2=t \\ \Rightarrow 2xdx=dt \\ \Rightarrow xdx=\frac{dt}{2} \\ \therefore I=\frac{1}{2}\int \sec t·dt \\ =\frac{1}{2}\log \left|\sec t+\tan t\right|+C \\ =\frac{1}{2}\log \left|\sec x^2+\tan x^2\right|+C\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\left(\mathit{\because }\mathit{}t\mathit{=}\mathit{}{x}^{\mathit{2}}\right) \end{gathered}$$