What is the integral of e^{2+x}cos(e^x+1) ?

The integral of e^{2+x}cos(e^x+1) is e^2sin(e^x+1)+C

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integral of e^{2+x}cos(e^x+1)

\int e^{2 + x} cos (e^{x} + 1) d x

e^{2} sin (e^{x} + 1) + C

Solution steps

\int e^{2 + x} cos (e^{x} + 1) d x

e^{2 + x} cos (e^{x} + 1) = e^{2} e^{x} cos (e^{x} + 1)

= \int e^{2} e^{x} cos (e^{x} + 1) d x

Take the constant out:

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

= e^{2} \cdot \int e^{x} cos (e^{x} + 1) d x

Apply u-substitution

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

Apply u-substitution

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

Use the common integral:

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

= e^{2} sin (v)

Substitute back

= e^{2} sin (e^{x} + 1)

Add a constant to the solution

= e^{2} sin (e^{x} + 1) + C

What is the integral of e^{2+x}cos(e^x+1) ?
The integral of e^{2+x}cos(e^x+1) is e^2sin(e^x+1)+C