What is the integral of e^{4x}cos(e^{4x}+pi) ?

The integral of e^{4x}cos(e^{4x}+pi) is -1/4 sin(e^{4x})+C

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integral of e^{4x}cos(e^{4x}+pi)

\int e^{4 x} cos (e^{4 x} + π) d x

- \frac{1}{4} sin (e^{4 x}) + C

Solution steps

\int e^{4 x} cos (e^{4 x} + π) d x

Rewrite using trig identities

= \int - e^{4 x} cos (e^{4 x}) d x

Take the constant out:

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

= - \int e^{4 x} cos (e^{4 x}) d x

Apply u-substitution

: \int e^{u} cos (e^{u}) \frac{1}{4} d u

= - \int e^{u} cos (e^{u}) \frac{1}{4} d u

Take the constant out:

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

= - \frac{1}{4} \cdot \int e^{u} cos (e^{u}) d u

Apply u-substitution

: \int cos (v) d v

= - \frac{1}{4} \cdot \int cos (v) d v

Use the common integral:

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

= - \frac{1}{4} sin (v)

Substitute back

= - \frac{1}{4} sin (e^{4 x})

Add a constant to the solution

= - \frac{1}{4} sin (e^{4 x}) + C

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What is the integral of e^{4x}cos(e^{4x}+pi) ?
The integral of e^{4x}cos(e^{4x}+pi) is -1/4 sin(e^{4x})+C