What is the integral of cos(3x)sin(x) ?

The integral of cos(3x)sin(x) is 1/2 (-1/4 cos(4x)+1/2 cos(2x))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of cos(3x)sin(x)

\int cos (3 x) sin (x) d x

\frac{1}{2} (- \frac{1}{4} cos (4 x) + \frac{1}{2} cos (2 x)) + C

Solution steps

\int cos (3 x) sin (x) d x

Use the following identity:

cos (t) sin (s) = \frac{sin ( s + t ) + sin ( s - t )}{2}

= \int \frac{sin ( x + 3 x ) + sin ( x - 3 x )}{2} d x

Take the constant out:

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

= \frac{1}{2} \cdot \int sin (x + 3 x) + sin (x - 3 x) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{2} (\int sin (x + 3 x) d x + \int sin (x - 3 x) d x)

\int sin (x + 3 x) d x = - \frac{1}{4} cos (4 x)

\int sin (x - 3 x) d x = \frac{1}{2} cos (2 x)

= \frac{1}{2} (- \frac{1}{4} cos (4 x) + \frac{1}{2} cos (2 x))

Add a constant to the solution

= \frac{1}{2} (- \frac{1}{4} cos (4 x) + \frac{1}{2} cos (2 x)) + C

(9 x^{6} + 4 x^{3})^{4}

8 \cdot e^{(- 2 t)} \cdot t \cdot cos (3 t) \cdot cos (t)

\frac{\partial}{\partial z} (\frac{z ^{2} + 2 x + y}{x y})

\int p (p + 9)^{8} d p

\int \frac{2 x - 3}{x} d x

What is the integral of cos(3x)sin(x) ?
The integral of cos(3x)sin(x) is 1/2 (-1/4 cos(4x)+1/2 cos(2x))+C