Question

Solve:
integral 4 sin x cos x e^2 cos X dx​

Answer 1

Integration

Solution:

Let, $cosx=z$ such that

$\quad sinx\:dx=-dz$

$\therefore \int 4\:sinx\:cosx\:e^{2\:cosx}\:dx$

$=-\int 4\:z\:e^{2z}\:dz$

$=-4\int z\:e^{2z}\:dz$

$=-4\:z\int e^{2z}\:dz+4\int \left[\frac{d}{dz}(z)\:\int e^{2z}\:dz\right]\:dz$

$=-4\:z\:\frac{e^{2z}}{2}+4\int \frac{e^{2z}}{2}\:dz$

$=-2\:z\:e^{2z}+e^{2z}+c$

$\quad$where $c=$ integral constant

$=-2\:cosx\:e^{2\:cosx}+e^{2\:cosx}+c$

This is the required integral.

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