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

The integral of cos(x/a)sin(x/a) is -a/4 cos((2x)/a)+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/a)sin(x/a)

\int cos (\frac{x}{a}) sin (\frac{x}{a}) d x

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

Solution steps

\int cos (\frac{x}{a}) sin (\frac{x}{a}) d x

Rewrite using trig identities

= \int \frac{1}{2} sin (\frac{2 x}{a}) 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{2 x}{a}) d x

Apply u-substitution

: \int \frac{a sin ( u )}{2} d u

= \frac{1}{2} \cdot \int \frac{a sin ( u )}{2} d u

Take the constant out:

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

= \frac{1}{2} \cdot \frac{a}{2} \cdot \int sin (u) d u

Use the common integral:

\int sin (u) d u = - cos (u)

= \frac{1}{2} \cdot \frac{a}{2} (- cos (u))

Substitute back

u = \frac{2 x}{a}

= \frac{1}{2} \cdot \frac{a}{2} (- cos (\frac{2 x}{a}))

Simplify

\frac{1}{2} \cdot \frac{a}{2} (- cos (\frac{2 x}{a})) : - \frac{a}{4} cos (\frac{2 x}{a})

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

Add a constant to the solution

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

\frac{d}{d x} (\frac{2 ^{x}}{1 - x ^{2}})

\int \frac{1}{x + !} d x

\frac{d}{d x} ((2 x - 1) (3 x + 4) (x + 7))

\frac{d}{d x} (\frac{y ^{2} - 3 x ^{2}}{y ^{4}})

\frac{\partial}{\partial x} (+ 6 - 5 + 6 x)

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