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

The integral of sin(2x)sqrt(cos(2x)) is -1/3 cos^{3/2}(2x)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

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

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

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

Solution steps

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

Apply u-substitution

= \int - u^{2} d u

Take the constant out:

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

= - \int u^{2} d u

Apply the Power Rule

= - \frac{u ^{3}}{3}

Substitute back

u = cos (2 x)

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

Simplify

- \frac{( cos ( 2 x ) ) ^{3}}{3} : - \frac{1}{3} cos^{\frac{3}{2}} (2 x)

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

Add a constant to the solution

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

What is the integral of sin(2x)sqrt(cos(2x)) ?
The integral of sin(2x)sqrt(cos(2x)) is -1/3 cos^{3/2}(2x)+C