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

The integral of cos(4x)e^{4x} is (e^{4x}sin(4x))/8+(e^{4x}cos(4x))/8+C

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

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

\frac{e ^{4 x} sin ( 4 x )}{8} + \frac{e ^{4 x} cos ( 4 x )}{8} + C

Solution steps

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

Apply Integration By Parts

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

Apply Integration By Parts

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

Take the constant out:

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

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

Therefore

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

Isolate

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

= \frac{e ^{4 x} sin ( 4 x )}{8} + \frac{e ^{4 x} cos ( 4 x )}{8}

Add a constant to the solution

= \frac{e ^{4 x} sin ( 4 x )}{8} + \frac{e ^{4 x} cos ( 4 x )}{8} + C

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What is the integral of cos(4x)e^{4x} ?
The integral of cos(4x)e^{4x} is (e^{4x}sin(4x))/8+(e^{4x}cos(4x))/8+C