What is the integral of cos(x)(e^{sin(x)}+sin^3(x)) ?

The integral of cos(x)(e^{sin(x)}+sin^3(x)) is e^{sin(x)}+(sin^4(x))/4+C

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integral of cos(x)(e^{sin(x)}+sin^3(x))

\int cos (x) (e^{s i n (x)} + sin^{3} (x)) d x

e^{s i n (x)} + \frac{sin ^{4} ( x )}{4} + C

Solution steps

\int cos (x) (e^{s i n (x)} + sin^{3} (x)) d x

Expand

cos (x) (e^{s i n (x)} + sin^{3} (x)) : e^{s i n (x)} cos (x) + sin^{3} (x) cos (x)

= \int e^{s i n (x)} cos (x) + sin^{3} (x) cos (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^{s i n (x)} cos (x) d x + \int sin^{3} (x) cos (x) d x

\int e^{s i n (x)} cos (x) d x = e^{s i n (x)}

\int sin^{3} (x) cos (x) d x = \frac{sin ^{4} ( x )}{4}

= e^{s i n (x)} + \frac{sin ^{4} ( x )}{4}

Add a constant to the solution

= e^{s i n (x)} + \frac{sin ^{4} ( x )}{4} + C

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What is the integral of cos(x)(e^{sin(x)}+sin^3(x)) ?
The integral of cos(x)(e^{sin(x)}+sin^3(x)) is e^{sin(x)}+(sin^4(x))/4+C