What is the integral of (e^{2cos(x)}+x)sin(x) ?

The integral of (e^{2cos(x)}+x)sin(x) is -1/2 e^{2cos(x)}-xcos(x)+sin(x)+C

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integral of (e^{2cos(x)}+x)sin(x)

\int (e^{2 c o s (x)} + x) sin (x) d x

- \frac{1}{2} e^{2 c o s (x)} - x cos (x) + sin (x) + C

Solution steps

\int (e^{2 c o s (x)} + x) sin (x) d x

Expand

(e^{2 c o s (x)} + x) sin (x) : e^{2 c o s (x)} sin (x) + x sin (x)

= \int e^{2 c o s (x)} sin (x) + x sin (x) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int e^{2 c o s (x)} sin (x) d x + \int x sin (x) d x

\int e^{2 c o s (x)} sin (x) d x = - \frac{1}{2} e^{2 c o s (x)}

\int x sin (x) d x = - x cos (x) + sin (x)

= - \frac{1}{2} e^{2 c o s (x)} - x cos (x) + sin (x)

Add a constant to the solution

= - \frac{1}{2} e^{2 c o s (x)} - x cos (x) + sin (x) + C

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What is the integral of (e^{2cos(x)}+x)sin(x) ?
The integral of (e^{2cos(x)}+x)sin(x) is -1/2 e^{2cos(x)}-xcos(x)+sin(x)+C