What is the integral of e^θsin(3θ) ?

The integral of e^θsin(3θ) is -(3e^θcos(3θ))/(10)+(e^θsin(3θ))/(10)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of e^θsin(3θ)

\int e^{θ} sin (3 θ) d θ

- \frac{3 e ^{θ} cos ( 3 θ )}{10} + \frac{e ^{θ} sin ( 3 θ )}{10} + C

Solution steps

\int e^{θ} sin (3 θ) d θ

Apply Integration By Parts

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

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

Apply Integration By Parts

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

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

Therefore

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

Isolate

\int e^{θ} sin (3 θ) d θ

= - \frac{3 e ^{θ} cos ( 3 θ )}{10} + \frac{e ^{θ} sin ( 3 θ )}{10}

Add a constant to the solution

= - \frac{3 e ^{θ} cos ( 3 θ )}{10} + \frac{e ^{θ} sin ( 3 θ )}{10} + C

\int 5 x^{3} e^{x^{4}} d x

\frac{d}{d t} (\frac{5}{5 - t})

x^{2} y^{2} = 4, (- 1, 2)

\int 4 - 25 x^{2} d x

\frac{d}{d x} (ln (\frac{x}{4}) + \frac{x - 4}{x})

What is the integral of e^θsin(3θ) ?
The integral of e^θsin(3θ) is -(3e^θcos(3θ))/(10)+(e^θsin(3θ))/(10)+C