What is the integral of-e^xsin(e^x) ?

The integral of-e^xsin(e^x) is cos(e^x)+C

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integral of-e^xsin(e^x)

\int - e^{x} sin (e^{x}) d x

cos (e^{x}) + C

Solution steps

\int - e^{x} sin (e^{x}) d x

Take the constant out:

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

= - \int e^{x} sin (e^{x}) d x

Apply u-substitution

= - \int sin (u) d u

Use the common integral:

\int sin (u) d u = - cos (u)

= - (- cos (u))

Substitute back

u = e^{x}

= - (- cos (e^{x}))

Simplify

= cos (e^{x})

Add a constant to the solution

= cos (e^{x}) + C