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

The integral of e^{2x}cos(e^{2x}+3) is 1/2 sin(e^{2x}+3)+C

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integral of e^{2x}cos(e^{2x}+3)

\int e^{2 x} cos (e^{2 x} + 3) d x

\frac{1}{2} sin (e^{2 x} + 3) + C

Solution steps

\int e^{2 x} cos (e^{2 x} + 3) d x

Apply u-substitution

: \int e^{u} cos (e^{u} + 3) \frac{1}{2} d u

= \int e^{u} cos (e^{u} + 3) \frac{1}{2} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int e^{u} cos (e^{u} + 3) d u

Apply u-substitution

: \int cos (v + 3) d v

= \frac{1}{2} \cdot \int cos (v + 3) d v

Apply u-substitution

: \int cos (w) d w

= \frac{1}{2} \cdot \int cos (w) d w

Use the common integral:

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

= \frac{1}{2} sin (w)

Substitute back

= \frac{1}{2} sin (e^{2 x} + 3)

Add a constant to the solution

= \frac{1}{2} sin (e^{2 x} + 3) + C

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What is the integral of e^{2x}cos(e^{2x}+3) ?
The integral of e^{2x}cos(e^{2x}+3) is 1/2 sin(e^{2x}+3)+C