What is the integral of 2e^xcos(e^x) ?

The integral of 2e^xcos(e^x) is 2sin(e^x)+C

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integral of 2e^xcos(e^x)

\int 2 e^{x} cos (e^{x}) d x

2 sin (e^{x}) + C

Solution steps

\int 2 e^{x} cos (e^{x}) d x

Take the constant out:

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

= 2 \cdot \int e^{x} cos (e^{x}) d x

Apply u-substitution

= 2 \cdot \int cos (u) d u

Use the common integral:

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

= 2 sin (u)

Substitute back

u = e^{x}

= 2 sin (e^{x})

Add a constant to the solution

= 2 sin (e^{x}) + C

What is the integral of 2e^xcos(e^x) ?
The integral of 2e^xcos(e^x) is 2sin(e^x)+C