What is the integral of cos^2(ax)sin(ax) ?

The integral of cos^2(ax)sin(ax) is -1/(3a)cos^3(ax)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of cos^2(ax)sin(ax)

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

- \frac{1}{3 a} cos^{3} (a x) + C

Solution steps

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

Apply u-substitution

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

= \int \frac{cos ^{2} ( u ) sin ( u )}{a} d u

Take the constant out:

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

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

Apply u-substitution

: \int - v^{2} d v

= \frac{1}{a} \cdot \int - v^{2} d v

Take the constant out:

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

= \frac{1}{a} (- \int v^{2} d v)

Apply the Power Rule

: \frac{v ^{3}}{3}

= \frac{1}{a} (- \frac{v ^{3}}{3})

Substitute back

= \frac{1}{a} (- \frac{cos ^{3} ( a x )}{3})

Simplify

\frac{1}{a} (- \frac{cos ^{3} ( a x )}{3}) : - \frac{1}{3 a} cos^{3} (a x)

= - \frac{1}{3 a} cos^{3} (a x)

Add a constant to the solution

= - \frac{1}{3 a} cos^{3} (a x) + C

\int \frac{1}{8 x ^{2}} d x

\int x x^{2} - 36 d x

\int \frac{( 5 - 2 x )}{12 x - 4 x ^{2}} d x

\int t 1 - t d t

\int \frac{6}{x ^{2}} cos (\frac{6}{x}) d x

What is the integral of cos^2(ax)sin(ax) ?
The integral of cos^2(ax)sin(ax) is -1/(3a)cos^3(ax)+C