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

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

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

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

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

Solution steps

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

Therefore

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

Isolate

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

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

Add a constant to the solution

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

f (x) = tan^{2} (3 e^{x})

\frac{7}{( 2 x + 1 ) ^{2}}

x^{^{''}} = - x + 0.1 x^{^{'}}

\frac{y ^{^{''}} + 10 y ^{^{'}} + 149}{4 y} = 0

x \to 2 lim (+ f (x))

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