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

The integral of cos^3(x)e^{sin(x)} is 2(-e^{sin(x)}+e^{sin(x)}sin(x))+e^{sin(x)}cos^2(x)+C

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

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

2 (- e^{s i n (x)} + e^{s i n (x)} sin (x)) + e^{s i n (x)} cos^{2} (x) + C

Solution steps

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

Apply u-substitution

: \int e^{u} (1 - u^{2}) d u

= \int e^{u} (1 - u^{2}) d u

Apply Integration By Parts

: e^{u} (1 - u^{2}) - \int - 2 e^{u} u d u

= e^{u} (1 - u^{2}) - \int - 2 e^{u} u d u

\int - 2 e^{u} u d u = - 2 (e^{u} u - e^{u})

= e^{u} (1 - u^{2}) - (- 2 (e^{u} u - e^{u}))

Substitute back

u = sin (x)

= e^{s i n (x)} (1 - sin^{2} (x)) - (- 2 (e^{s i n (x)} sin (x) - e^{s i n (x)}))

Simplify

e^{s i n (x)} (1 - sin^{2} (x)) - (- 2 (e^{s i n (x)} sin (x) - e^{s i n (x)})) : 2 (- e^{s i n (x)} + e^{s i n (x)} sin (x)) + e^{s i n (x)} cos^{2} (x)

= 2 (- e^{s i n (x)} + e^{s i n (x)} sin (x)) + e^{s i n (x)} cos^{2} (x)

Add a constant to the solution

= 2 (- e^{s i n (x)} + e^{s i n (x)} sin (x)) + e^{s i n (x)} cos^{2} (x) + C

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What is the integral of cos^3(x)e^{sin(x)} ?
The integral of cos^3(x)e^{sin(x)} is 2(-e^{sin(x)}+e^{sin(x)}sin(x))+e^{sin(x)}cos^2(x)+C