en

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of sin^{12}(x)

Solution

\int sin^{12} (x) d x

Solution

- \frac{cos ( x ) sin ^{11} ( x )}{12} + \frac{11}{12} (- \frac{cos ( x ) sin ^{9} ( x )}{10} + \frac{9}{10} (- \frac{cos ( x ) sin ^{7} ( x )}{8} + \frac{7}{8} (- \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^{12} (x) d x

Apply Integral Reduction

= - \frac{cos ( x ) sin ^{11} ( x )}{12} + \frac{11}{12} \cdot \int sin^{10} (x) d x

\int sin^{10} (x) d x = - \frac{cos ( x ) sin ^{9} ( x )}{10} + \frac{9}{10} (- \frac{cos ( x ) sin ^{7} ( x )}{8} + \frac{7}{8} (- \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)))))

= - \frac{cos ( x ) sin ^{11} ( x )}{12} + \frac{11}{12} (- \frac{cos ( x ) sin ^{9} ( x )}{10} + \frac{9}{10} (- \frac{cos ( x ) sin ^{7} ( x )}{8} + \frac{7}{8} (- \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 ^{11} ( x )}{12} + \frac{11}{12} (- \frac{cos ( x ) sin ^{9} ( x )}{10} + \frac{9}{10} (- \frac{cos ( x ) sin ^{7} ( x )}{8} + \frac{7}{8} (- \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

Popular Examples

derivative of x-5sin(x)

\frac{d}{d x} (x - 5 sin (x))

(d^2)/(dx^2)(sqrt(x+10))

\frac{d ^{2}}{d x ^{2}} (x + 10)

derivative of x^{4/x}

\frac{d}{d x} (x^{\frac{4}{x}})

y^{^{'}} = x (7 - y)

limit as x approaches 0 of 5(x-3)ln(x-3)

x \to 0 lim (5 (x - 3) ln (x - 3))

Frequently Asked Questions (FAQ)

What is the integral of sin^{12}(x) ?
The integral of sin^{12}(x) is