What is the integral of e^{2θ}cos(3θ) ?

The integral of e^{2θ}cos(3θ) is (3e^{2θ}sin(3θ))/(13)+(2e^{2θ}cos(3θ))/(13)+C

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integral of e^{2θ}cos(3θ)

\int e^{2 θ} cos (3 θ) d θ

\frac{3 e ^{2 θ} sin ( 3 θ )}{13} + \frac{2 e ^{2 θ} cos ( 3 θ )}{13} + C

Solution steps

\int e^{2 θ} cos (3 θ) d θ

Apply Integration By Parts

: \frac{1}{3} e^{2 θ} sin (3 θ) - \int \frac{2}{3} e^{2 θ} sin (3 θ) d θ

= \frac{1}{3} e^{2 θ} sin (3 θ) - \int \frac{2}{3} e^{2 θ} sin (3 θ) d θ

Take the constant out:

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

= \frac{1}{3} e^{2 θ} sin (3 θ) - \frac{2}{3} \cdot \int e^{2 θ} sin (3 θ) d θ

Apply Integration By Parts

: - \frac{1}{3} e^{2 θ} cos (3 θ) - \int - \frac{2}{3} e^{2 θ} cos (3 θ) d θ

= \frac{1}{3} e^{2 θ} sin (3 θ) - \frac{2}{3} (- \frac{1}{3} e^{2 θ} cos (3 θ) - \int - \frac{2}{3} e^{2 θ} cos (3 θ) d θ)

Take the constant out:

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

= \frac{1}{3} e^{2 θ} sin (3 θ) - \frac{2}{3} (- \frac{1}{3} e^{2 θ} cos (3 θ) - (- \frac{2}{3} \cdot \int e^{2 θ} cos (3 θ) d θ))

Therefore

\int e^{2 θ} cos (3 θ) d θ = \frac{1}{3} e^{2 θ} sin (3 θ) - \frac{2}{3} (- \frac{1}{3} e^{2 θ} cos (3 θ) - (- \frac{2}{3} \cdot \int e^{2 θ} cos (3 θ) d θ))

Isolate

\int e^{2 θ} cos (3 θ) d θ

= \frac{3 e ^{2 θ} sin ( 3 θ )}{13} + \frac{2 e ^{2 θ} cos ( 3 θ )}{13}

Add a constant to the solution

= \frac{3 e ^{2 θ} sin ( 3 θ )}{13} + \frac{2 e ^{2 θ} cos ( 3 θ )}{13} + C

y = 2 x^{2} - 12 x + 6

\int \frac{1}{x - x ^{2}} d x

\frac{d}{d x} (π \cdot x^{2})

y = \frac{( 3 x ^{5} + 9 )}{x ^{3}}

\frac{d}{d s} (ln (s^{2} + 9))

What is the integral of e^{2θ}cos(3θ) ?
The integral of e^{2θ}cos(3θ) is (3e^{2θ}sin(3θ))/(13)+(2e^{2θ}cos(3θ))/(13)+C