What is the integral of e^6θsin(7θ) ?

The integral of e^6θsin(7θ) is e^6(-1/7 θcos(7θ)+1/49 sin(7θ))+C

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integral of e^6θsin(7θ)

\int e^{6} θ sin (7 θ) d θ

e^{6} (- \frac{1}{7} θ cos (7 θ) + \frac{1}{49} sin (7 θ)) + C

Solution steps

\int 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

= e^{6} \cdot \int θ sin (7 θ) d θ

Apply Integration By Parts

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

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

\int - \frac{1}{7} cos (7 θ) d θ = - \frac{1}{49} sin (7 θ)

= e^{6} (- \frac{1}{7} θ cos (7 θ) - (- \frac{1}{49} sin (7 θ)))

Simplify

= e^{6} (- \frac{1}{7} θ cos (7 θ) + \frac{1}{49} sin (7 θ))

Add a constant to the solution

= e^{6} (- \frac{1}{7} θ cos (7 θ) + \frac{1}{49} sin (7 θ)) + C

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What is the integral of e^6θsin(7θ) ?
The integral of e^6θsin(7θ) is e^6(-1/7 θcos(7θ)+1/49 sin(7θ))+C