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

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

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

\int sin^{3} (x) e^{c o s (x)} d x

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

Solution steps

\int sin^{3} (x) e^{c o s (x)} d x

Apply Trigonometric Substitution

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Apply Integration By Parts

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

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

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

Substitute back

u = cos (x)

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

Simplify

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

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

Add a constant to the solution

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

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