What is the integral of cos(x/3)sin(x/2) ?

The integral of cos(x/3)sin(x/2) is 1/2 (-6/5 cos((5x)/6)-6cos(x/6))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of cos(x/3)sin(x/2)

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

\frac{1}{2} (- \frac{6}{5} cos (\frac{5 x}{6}) - 6 cos (\frac{x}{6})) + C

Solution steps

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

Use the following identity:

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

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

\int sin (\frac{x}{2} + \frac{x}{3}) d x = - \frac{6}{5} cos (\frac{5 x}{6})

\int sin (\frac{x}{2} - \frac{x}{3}) d x = - 6 cos (\frac{x}{6})

= \frac{1}{2} (- \frac{6}{5} cos (\frac{5 x}{6}) - 6 cos (\frac{x}{6}))

Add a constant to the solution

= \frac{1}{2} (- \frac{6}{5} cos (\frac{5 x}{6}) - 6 cos (\frac{x}{6})) + C

\int \frac{a}{1 - cos ( b x )} d x

\int 200 - 0.02 x d x

\int x \cdot sec^{2} (4 x) d x

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

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

What is the integral of cos(x/3)sin(x/2) ?
The integral of cos(x/3)sin(x/2) is 1/2 (-6/5 cos((5x)/6)-6cos(x/6))+C