What is the integral of e^{7x}sin(x) ?

The integral of e^{7x}sin(x) is -(e^{7x}cos(x))/(50)+(7e^{7x}sin(x))/(50)+C

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integral of e^{7x}sin(x)

\int e^{7 x} sin (x) d x

- \frac{e ^{7 x} cos ( x )}{50} + \frac{7 e ^{7 x} sin ( x )}{50} + C

Solution steps

\int e^{7 x} sin (x) d x

Apply Integration By Parts

= - e^{7 x} cos (x) - \int - 7 e^{7 x} cos (x) d x

Take the constant out:

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

= - e^{7 x} cos (x) - (- 7 \cdot \int e^{7 x} cos (x) d x)

Apply Integration By Parts

= - e^{7 x} cos (x) - (- 7 (e^{7 x} sin (x) - \int e^{7 x} \cdot 7 sin (x) d x))

Take the constant out:

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

= - e^{7 x} cos (x) - (- 7 (e^{7 x} sin (x) - 7 \cdot \int e^{7 x} sin (x) d x))

Therefore

\int e^{7 x} sin (x) d x = - e^{7 x} cos (x) - (- 7 (e^{7 x} sin (x) - 7 \cdot \int e^{7 x} sin (x) d x))

Isolate

\int e^{7 x} sin (x) d x

= - \frac{e ^{7 x} cos ( x )}{50} + \frac{7 e ^{7 x} sin ( x )}{50}

Add a constant to the solution

= - \frac{e ^{7 x} cos ( x )}{50} + \frac{7 e ^{7 x} sin ( x )}{50} + C

What is the integral of e^{7x}sin(x) ?
The integral of e^{7x}sin(x) is -(e^{7x}cos(x))/(50)+(7e^{7x}sin(x))/(50)+C