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

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

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

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

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

Solution steps

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

Apply Integration By Parts

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

Take the constant out:

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

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

Apply Integration By Parts

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

Take the constant out:

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

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

Therefore

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

Isolate

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

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

Add a constant to the solution

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

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