What is the integral of+e^{7θ}sin(8θ) ?

The integral of+e^{7θ}sin(8θ) is -(8e^{7θ}cos(8θ))/(113)+(7e^{7θ}sin(8θ))/(113)+C

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integral of+e^{7θ}sin(8θ)

\int + e^{7 θ} sin (8 θ) d θ

- \frac{8 e ^{7 θ} cos ( 8 θ )}{113} + \frac{7 e ^{7 θ} sin ( 8 θ )}{113} + C

Solution steps

\int e^{7 θ} sin (8 θ) d θ

Apply Integration By Parts

: - \frac{1}{8} e^{7 θ} cos (8 θ) - \int - \frac{7}{8} e^{7 θ} cos (8 θ) d θ

= - \frac{1}{8} e^{7 θ} cos (8 θ) - \int - \frac{7}{8} e^{7 θ} cos (8 θ) d θ

Take the constant out:

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

= - \frac{1}{8} e^{7 θ} cos (8 θ) - (- \frac{7}{8} \cdot \int e^{7 θ} cos (8 θ) d θ)

Apply Integration By Parts

: \frac{1}{8} e^{7 θ} sin (8 θ) - \int \frac{7}{8} e^{7 θ} sin (8 θ) d θ

= - \frac{1}{8} e^{7 θ} cos (8 θ) - (- \frac{7}{8} (\frac{1}{8} e^{7 θ} sin (8 θ) - \int \frac{7}{8} e^{7 θ} sin (8 θ) d θ))

Take the constant out:

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

= - \frac{1}{8} e^{7 θ} cos (8 θ) - (- \frac{7}{8} (\frac{1}{8} e^{7 θ} sin (8 θ) - \frac{7}{8} \cdot \int e^{7 θ} sin (8 θ) d θ))

Therefore

\int e^{7 θ} sin (8 θ) d θ = - \frac{1}{8} e^{7 θ} cos (8 θ) - (- \frac{7}{8} (\frac{1}{8} e^{7 θ} sin (8 θ) - \frac{7}{8} \cdot \int e^{7 θ} sin (8 θ) d θ))

Isolate

\int e^{7 θ} sin (8 θ) d θ

= - \frac{8 e ^{7 θ} cos ( 8 θ )}{113} + \frac{7 e ^{7 θ} sin ( 8 θ )}{113}

Add a constant to the solution

= - \frac{8 e ^{7 θ} cos ( 8 θ )}{113} + \frac{7 e ^{7 θ} sin ( 8 θ )}{113} + C

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What is the integral of+e^{7θ}sin(8θ) ?
The integral of+e^{7θ}sin(8θ) is -(8e^{7θ}cos(8θ))/(113)+(7e^{7θ}sin(8θ))/(113)+C