What is the integral of e^{9x}cos(8x) ?

The integral of e^{9x}cos(8x) is (8e^{9x}sin(8x))/(145)+(9e^{9x}cos(8x))/(145)+C

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integral of e^{9x}cos(8x)

\int e^{9 x} cos (8 x) d x

\frac{8 e ^{9 x} sin ( 8 x )}{145} + \frac{9 e ^{9 x} cos ( 8 x )}{145} + C

Solution steps

\int e^{9 x} cos (8 x) d x

Apply Integration By Parts

: \frac{1}{8} e^{9 x} sin (8 x) - \int \frac{9}{8} e^{9 x} sin (8 x) d x

= \frac{1}{8} e^{9 x} sin (8 x) - \int \frac{9}{8} e^{9 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^{9 x} sin (8 x) - \frac{9}{8} \cdot \int e^{9 x} sin (8 x) d x

Apply Integration By Parts

: - \frac{1}{8} e^{9 x} cos (8 x) - \int - \frac{9}{8} e^{9 x} cos (8 x) d x

= \frac{1}{8} e^{9 x} sin (8 x) - \frac{9}{8} (- \frac{1}{8} e^{9 x} cos (8 x) - \int - \frac{9}{8} e^{9 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^{9 x} sin (8 x) - \frac{9}{8} (- \frac{1}{8} e^{9 x} cos (8 x) - (- \frac{9}{8} \cdot \int e^{9 x} cos (8 x) d x))

Therefore

\int e^{9 x} cos (8 x) d x = \frac{1}{8} e^{9 x} sin (8 x) - \frac{9}{8} (- \frac{1}{8} e^{9 x} cos (8 x) - (- \frac{9}{8} \cdot \int e^{9 x} cos (8 x) d x))

Isolate

\int e^{9 x} cos (8 x) d x

= \frac{8 e ^{9 x} sin ( 8 x )}{145} + \frac{9 e ^{9 x} cos ( 8 x )}{145}

Add a constant to the solution

= \frac{8 e ^{9 x} sin ( 8 x )}{145} + \frac{9 e ^{9 x} cos ( 8 x )}{145} + C

\frac{\partial}{\partial z} (2 x^{\frac{y}{z}})

\int 4 x e^{- x} d x

y = 4 x

\frac{\partial}{\partial x} (x^{3} y^{2} - 4 y x^{2} + 3 y x - 4)

f (x) = \frac{x}{x ^{2} - 9}

What is the integral of e^{9x}cos(8x) ?
The integral of e^{9x}cos(8x) is (8e^{9x}sin(8x))/(145)+(9e^{9x}cos(8x))/(145)+C