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  4. The value of the definite integral ∫ limits0π((1-x sin 2 x) e cos 2 x+(1+x sin 2 x) e sin 2 x) d x is equal to

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Q. The value of the definite integral $\int\limits_0^\pi\left((1-x \sin 2 x) e^{\cos ^2 x}+(1+x \sin 2 x) e^{\sin ^2 x}\right) d x$ is equal to

Integrals

Solution:

$\int\left(f(x)+x f^{\prime}(x)\right) d x=x f(x)+C$
So, $\left.\quad I=x\left(e^{\cos ^2 x}+e^{\sin ^2 x}\right)\right]_0^\pi=\pi(e+1)$.