What is the integral of cos(2x)sin(5x) ?

The integral of cos(2x)sin(5x) is 1/2 (-1/7 cos(7x)-1/3 cos(3x))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of cos(2x)sin(5x)

\int cos (2 x) sin (5 x) d x

\frac{1}{2} (- \frac{1}{7} cos (7 x) - \frac{1}{3} cos (3 x)) + C

Solution steps

\int cos (2 x) sin (5 x) d x

Use the following identity:

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

= \int \frac{sin ( 5 x + 2 x ) + sin ( 5 x - 2 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 (5 x + 2 x) + sin (5 x - 2 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 (5 x + 2 x) d x + \int sin (5 x - 2 x) d x)

\int sin (5 x + 2 x) d x = - \frac{1}{7} cos (7 x)

\int sin (5 x - 2 x) d x = - \frac{1}{3} cos (3 x)

= \frac{1}{2} (- \frac{1}{7} cos (7 x) - \frac{1}{3} cos (3 x))

Add a constant to the solution

= \frac{1}{2} (- \frac{1}{7} cos (7 x) - \frac{1}{3} cos (3 x)) + C

\int (x^{2} - 3 x + 2) e^{2 x} d x

\int \frac{x ln ( x )}{( 1 + x ^{2} ) ^{2}} d x

\int \frac{cos ( 2 x )}{sin ( x ) + cos ( x )} d x

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

What is the integral of cos(2x)sin(5x) ?
The integral of cos(2x)sin(5x) is 1/2 (-1/7 cos(7x)-1/3 cos(3x))+C