What is the integral of 8e^{4x}cos(6x) ?

The integral of 8e^{4x}cos(6x) is 8((3e^{4x}sin(6x))/(26)+(e^{4x}cos(6x))/(13))+C

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integral of 8e^{4x}cos(6x)

\int 8 e^{4 x} cos (6 x) d x

8 (\frac{3 e ^{4 x} sin ( 6 x )}{26} + \frac{e ^{4 x} cos ( 6 x )}{13}) + C

Solution steps

\int 8 e^{4 x} cos (6 x) d x

Take the constant out:

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

= 8 \cdot \int e^{4 x} cos (6 x) d x

Apply Integration By Parts

: \frac{1}{6} e^{4 x} sin (6 x) - \int \frac{2}{3} e^{4 x} sin (6 x) d x

= 8 (\frac{1}{6} e^{4 x} sin (6 x) - \int \frac{2}{3} e^{4 x} sin (6 x) d x)

Take the constant out:

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

= 8 (\frac{1}{6} e^{4 x} sin (6 x) - \frac{2}{3} \cdot \int e^{4 x} sin (6 x) d x)

Apply Integration By Parts

: - \frac{1}{6} e^{4 x} cos (6 x) - \int - \frac{2}{3} e^{4 x} cos (6 x) d x

= 8 (\frac{1}{6} e^{4 x} sin (6 x) - \frac{2}{3} (- \frac{1}{6} e^{4 x} cos (6 x) - \int - \frac{2}{3} e^{4 x} cos (6 x) d x))

Take the constant out:

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

= 8 (\frac{1}{6} e^{4 x} sin (6 x) - \frac{2}{3} (- \frac{1}{6} e^{4 x} cos (6 x) - (- \frac{2}{3} \cdot \int e^{4 x} cos (6 x) d x)))

Therefore

8 \cdot \int e^{4 x} cos (6 x) d x = 8 (\frac{1}{6} e^{4 x} sin (6 x) - \frac{2}{3} (- \frac{1}{6} e^{4 x} cos (6 x) - (- \frac{2}{3} \cdot \int e^{4 x} cos (6 x) d x)))

Isolate

\int e^{4 x} cos (6 x) d x

= 8 (\frac{3 e ^{4 x} sin ( 6 x )}{26} + \frac{e ^{4 x} cos ( 6 x )}{13})

Add a constant to the solution

= 8 (\frac{3 e ^{4 x} sin ( 6 x )}{26} + \frac{e ^{4 x} cos ( 6 x )}{13}) + C

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What is the integral of 8e^{4x}cos(6x) ?
The integral of 8e^{4x}cos(6x) is 8((3e^{4x}sin(6x))/(26)+(e^{4x}cos(6x))/(13))+C