What is the integral of e^{6θ}sin(7θ) ?

The integral of e^{6θ}sin(7θ) is -(7e^{6θ}cos(7θ))/(85)+(6e^{6θ}sin(7θ))/(85)+C

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integral of e^{6θ}sin(7θ)

\int e^{6 θ} sin (7 θ) d θ

- \frac{7 e ^{6 θ} cos ( 7 θ )}{85} + \frac{6 e ^{6 θ} sin ( 7 θ )}{85} + C

Solution steps

\int e^{6 θ} sin (7 θ) d θ

Apply Integration By Parts

= - \frac{1}{7} e^{6 θ} cos (7 θ) - \int - \frac{6}{7} e^{6 θ} cos (7 θ) d θ

Take the constant out:

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

= - \frac{1}{7} e^{6 θ} cos (7 θ) - (- \frac{6}{7} \cdot \int e^{6 θ} cos (7 θ) d θ)

Apply Integration By Parts

= - \frac{1}{7} e^{6 θ} cos (7 θ) - (- \frac{6}{7} (\frac{1}{7} e^{6 θ} sin (7 θ) - \int \frac{6}{7} e^{6 θ} sin (7 θ) d θ))

Take the constant out:

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

= - \frac{1}{7} e^{6 θ} cos (7 θ) - (- \frac{6}{7} (\frac{1}{7} e^{6 θ} sin (7 θ) - \frac{6}{7} \cdot \int e^{6 θ} sin (7 θ) d θ))

Therefore

\int e^{6 θ} sin (7 θ) d θ = - \frac{1}{7} e^{6 θ} cos (7 θ) - (- \frac{6}{7} (\frac{1}{7} e^{6 θ} sin (7 θ) - \frac{6}{7} \cdot \int e^{6 θ} sin (7 θ) d θ))

Isolate

\int e^{6 θ} sin (7 θ) d θ

= - \frac{7 e ^{6 θ} cos ( 7 θ )}{85} + \frac{6 e ^{6 θ} sin ( 7 θ )}{85}

Add a constant to the solution

= - \frac{7 e ^{6 θ} cos ( 7 θ )}{85} + \frac{6 e ^{6 θ} sin ( 7 θ )}{85} + C

What is the integral of e^{6θ}sin(7θ) ?
The integral of e^{6θ}sin(7θ) is -(7e^{6θ}cos(7θ))/(85)+(6e^{6θ}sin(7θ))/(85)+C