Question

The value of the definite integral ${\int }_0^{\sqrt{\ln (\frac{\pi }{2})}}\cos \left(e^{x^2}\right)2x{e}^{x^2}dx$ is:

• A. $1$
• B. $1+\left(\sin 1\right)$
• C. $1-\left(\sin 1\right)$
• D. $\left(\sin 1\right)-1$

Answer 1

The correct option is C $1-\left(\sin 1\right)$

Let, $z=e^{x^2}⟹dz=2x{e}^{x^2}dx$

When $x=0,z=1$ and when $x=\sqrt{\ln \left(\frac{\pi }{2}\right)},z=\frac{\pi }{2}$

Hence, integration becomes:-

${\int }_1^{\pi /2}\cos zdz$

$={\left[\sin z\right]}_1^{\pi /2}$

$=1-\left(\sin 1\right)$

Hence, the answer is option-(C).