Question

The value of the definite integral $\underset{0}{\overset{\sqrt{\ln \left(\frac{\pi }{2}\right)}}{\int }}2x{e}^{x^2}\cos \left(e^{x^2}\right)dx$

• A. $1$
• B. $1+\sin 1$
• C. $1-\sin 1$
• D. $\sin 1-1$

Answer 1

The correct option is C $1-\sin 1$

$I=\underset{0}{\overset{\sqrt{\ln \left(\frac{\pi }{2}\right)}}{\int }}2x{e}^{x^2}\cos \left(e^{x^2}\right)dx$
Put $e^{x^2}=t\Rightarrow e^{x^2}2xdx=dt$
$\Rightarrow I=\underset{1}{\overset{\pi /2}{\int }}\cos tdt\Rightarrow I={\left[\sin t\right]}_1^{\pi /2}=1-\sin 1$