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

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

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

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

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

Solution steps

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

Apply Integration By Parts

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

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

Take the constant out:

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

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

Therefore

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

Isolate

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

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

Add a constant to the solution

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

\int I n (4 x) d x

f (x) = \frac{2}{x + 1}, at x = 3, y = 1

x - 2 x

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

3 x y + y^{2} + (x^{2} + x y) y^{^{'}} = 0

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