What is the integral of e^{5x}sin(4x) ?

The integral of e^{5x}sin(4x) is -(4e^{5x}cos(4x))/(41)+(5e^{5x}sin(4x))/(41)+C

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integral of e^{5x}sin(4x)

\int e^{5 x} sin (4 x) d x

- \frac{4 e ^{5 x} cos ( 4 x )}{41} + \frac{5 e ^{5 x} sin ( 4 x )}{41} + C

Solution steps

\int e^{5 x} sin (4 x) d x

Apply Integration By Parts

= - \frac{1}{4} e^{5 x} cos (4 x) - \int - \frac{5}{4} e^{5 x} cos (4 x) d x

Take the constant out:

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

= - \frac{1}{4} e^{5 x} cos (4 x) - (- \frac{5}{4} \cdot \int e^{5 x} cos (4 x) d x)

Apply Integration By Parts

= - \frac{1}{4} e^{5 x} cos (4 x) - (- \frac{5}{4} (\frac{1}{4} e^{5 x} sin (4 x) - \int \frac{5}{4} e^{5 x} sin (4 x) d x))

Take the constant out:

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

= - \frac{1}{4} e^{5 x} cos (4 x) - (- \frac{5}{4} (\frac{1}{4} e^{5 x} sin (4 x) - \frac{5}{4} \cdot \int e^{5 x} sin (4 x) d x))

Therefore

\int e^{5 x} sin (4 x) d x = - \frac{1}{4} e^{5 x} cos (4 x) - (- \frac{5}{4} (\frac{1}{4} e^{5 x} sin (4 x) - \frac{5}{4} \cdot \int e^{5 x} sin (4 x) d x))

Isolate

\int e^{5 x} sin (4 x) d x

= - \frac{4 e ^{5 x} cos ( 4 x )}{41} + \frac{5 e ^{5 x} sin ( 4 x )}{41}

Add a constant to the solution

= - \frac{4 e ^{5 x} cos ( 4 x )}{41} + \frac{5 e ^{5 x} sin ( 4 x )}{41} + C

What is the integral of e^{5x}sin(4x) ?
The integral of e^{5x}sin(4x) is -(4e^{5x}cos(4x))/(41)+(5e^{5x}sin(4x))/(41)+C