Integral of exponential function

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$\int \left(\cos5x\right) \; e^{\sin5x} \; dx$

**Answer**

$\int \left(\cos5x\right) \; e^{\sin5x} \; dx$

Substitute $u= \sin5x$. So, $du =5\cos5x\; dx$

And, $\frac{1}{5}\; du = \cos 5x\; dx$

Then $\int \left(\cos5x\right)\;e^{\sin5x} \; dx$

$=\frac{1}{5}\int e^{u} \; du$

$=\frac{1}{5}e^u + C $

Substituting back $u= \sin 5x$

$=\frac{1}{5}e^{\sin 5x} + C$