What is the integral of e^{8x}cos(5x) ?

The integral of e^{8x}cos(5x) is (5e^{8x}sin(5x))/(89)+(8e^{8x}cos(5x))/(89)+C

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integral of e^{8x}cos(5x)

\int e^{8 x} cos (5 x) d x

\frac{5 e ^{8 x} sin ( 5 x )}{89} + \frac{8 e ^{8 x} cos ( 5 x )}{89} + C

Solution steps

\int e^{8 x} cos (5 x) d x

Apply Integration By Parts

= \frac{1}{5} e^{8 x} sin (5 x) - \int \frac{8}{5} e^{8 x} sin (5 x) d x

Take the constant out:

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

= \frac{1}{5} e^{8 x} sin (5 x) - \frac{8}{5} \cdot \int e^{8 x} sin (5 x) d x

Apply Integration By Parts

= \frac{1}{5} e^{8 x} sin (5 x) - \frac{8}{5} (- \frac{1}{5} e^{8 x} cos (5 x) - \int - \frac{8}{5} e^{8 x} cos (5 x) d x)

Take the constant out:

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

= \frac{1}{5} e^{8 x} sin (5 x) - \frac{8}{5} (- \frac{1}{5} e^{8 x} cos (5 x) - (- \frac{8}{5} \cdot \int e^{8 x} cos (5 x) d x))

Therefore

\int e^{8 x} cos (5 x) d x = \frac{1}{5} e^{8 x} sin (5 x) - \frac{8}{5} (- \frac{1}{5} e^{8 x} cos (5 x) - (- \frac{8}{5} \cdot \int e^{8 x} cos (5 x) d x))

Isolate

\int e^{8 x} cos (5 x) d x

= \frac{5 e ^{8 x} sin ( 5 x )}{89} + \frac{8 e ^{8 x} cos ( 5 x )}{89}

Add a constant to the solution

= \frac{5 e ^{8 x} sin ( 5 x )}{89} + \frac{8 e ^{8 x} cos ( 5 x )}{89} + C

What is the integral of e^{8x}cos(5x) ?
The integral of e^{8x}cos(5x) is (5e^{8x}sin(5x))/(89)+(8e^{8x}cos(5x))/(89)+C