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

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

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

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

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

Solution steps

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

Apply u-substitution

: \int - e^{u} u d u

= \int - e^{u} u d u

Take the constant out:

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

= - \int e^{u} u d u

Apply Integration By Parts

: e^{u} u - \int e^{u} d u

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

\int e^{u} d u = e^{u}

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

Substitute back

u = cos (x)

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

Simplify

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

Add a constant to the solution

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

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