What is the integral of e^{-θ}cos(7θ) ?

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

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integral of e^{-θ}cos(7θ)

\int e^{- θ} cos (7 θ) d θ

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

Solution steps

\int e^{- θ} cos (7 θ) d θ

Apply Integration By Parts

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

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

Apply Integration By Parts

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

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

Therefore

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

Isolate

\int e^{- θ} cos (7 θ) d θ

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

Add a constant to the solution

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

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