What is the integral of e^{8x}cos(4x) ?

The integral of e^{8x}cos(4x) is (e^{8x}sin(4x))/(20)+(e^{8x}cos(4x))/(10)+C

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integral of e^{8x}cos(4x)

\int e^{8 x} cos (4 x) d x

\frac{e ^{8 x} sin ( 4 x )}{20} + \frac{e ^{8 x} cos ( 4 x )}{10} + C

Solution steps

\int e^{8 x} cos (4 x) d x

Apply Integration By Parts

: \frac{1}{4} e^{8 x} sin (4 x) - \int 2 e^{8 x} sin (4 x) d x

= \frac{1}{4} e^{8 x} sin (4 x) - \int 2 e^{8 x} sin (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^{8 x} sin (4 x) - 2 \cdot \int e^{8 x} sin (4 x) d x

Apply Integration By Parts

: - \frac{1}{4} e^{8 x} cos (4 x) - \int - 2 e^{8 x} cos (4 x) d x

= \frac{1}{4} e^{8 x} sin (4 x) - 2 (- \frac{1}{4} e^{8 x} cos (4 x) - \int - 2 e^{8 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^{8 x} sin (4 x) - 2 (- \frac{1}{4} e^{8 x} cos (4 x) - (- 2 \cdot \int e^{8 x} cos (4 x) d x))

Therefore

\int e^{8 x} cos (4 x) d x = \frac{1}{4} e^{8 x} sin (4 x) - 2 (- \frac{1}{4} e^{8 x} cos (4 x) - (- 2 \cdot \int e^{8 x} cos (4 x) d x))

Isolate

\int e^{8 x} cos (4 x) d x

= \frac{e ^{8 x} sin ( 4 x )}{20} + \frac{e ^{8 x} cos ( 4 x )}{10}

Add a constant to the solution

= \frac{e ^{8 x} sin ( 4 x )}{20} + \frac{e ^{8 x} cos ( 4 x )}{10} + C

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What is the integral of e^{8x}cos(4x) ?
The integral of e^{8x}cos(4x) is (e^{8x}sin(4x))/(20)+(e^{8x}cos(4x))/(10)+C