What is the integral of (sin(x))^6 ?

The integral of (sin(x))^6 is -(cos(x)sin^5(x))/6+5/6 (-1/4 sin^3(x)cos(x)+3/8 (x-1/2 sin(2x)))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (sin(x))^6

\int (sin (x))^{6} d x

- \frac{cos ( x ) sin ^{5} ( x )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{2} sin (2 x))) + C

Solution steps

\int (sin (x))^{6} d x

Simplify

= \int sin^{6} (x) d x

Apply Integral Reduction

= - \frac{cos ( x ) sin ^{5} ( x )}{6} + \frac{5}{6} \cdot \int sin^{4} (x) d x

\int sin^{4} (x) d x = - \frac{1}{4} sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{2} sin (2 x))

= - \frac{cos ( x ) sin ^{5} ( x )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{2} sin (2 x)))

Add a constant to the solution

= - \frac{cos ( x ) sin ^{5} ( x )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{2} sin (2 x))) + C

\frac{\partial}{\partial x} (z e^{x})

\int 4 e^{x} + cos (\frac{x}{2}) d x

\frac{d}{d x} ((- 25 x^{2} + 1)^{2})

\frac{\partial}{\partial x} (- \frac{7 x ^{2}}{( x - y ) ^{2}})

\frac{d}{d x} (a x^{b^{2} - c})

What is the integral of (sin(x))^6 ?
The integral of (sin(x))^6 is -(cos(x)sin^5(x))/6+5/6 (-1/4 sin^3(x)cos(x)+3/8 (x-1/2 sin(2x)))+C