en

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of sin^6(x/4)

Solution

\int sin^{6} (\frac{x}{4}) d x

Solution

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

Solution steps

\int sin^{6} (\frac{x}{4}) d x

Apply u-substitution

: \int sin^{6} (u) \cdot 4 d u

= \int sin^{6} (u) \cdot 4 d u

Take the constant out:

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

= 4 \cdot \int sin^{6} (u) d u

Apply Integral Reduction

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

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

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

Substitute back

u = \frac{x}{4}

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

Simplify

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

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

Add a constant to the solution

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

Graph

Popular Examples

integral of 1/(sqrt(5-x^2-2x))

\int \frac{1}{5 - x ^{2} - 2 x} d x

integral of (9x^3-8x+4)/(x^3-x^2)

\int \frac{9 x ^{3} - 8 x + 4}{x ^{3} - x ^{2}} d x

integral of ((x^2+1)/(x^3+3x+1)+5)

\int (\frac{x ^{2} + 1}{x ^{3} + 3 x + 1} + 5) d x

integral of (2x-1)/(x-2)

\int \frac{2 x - 1}{x - 2} d x

integral of 1/((100-x)^2)

\int \frac{1}{( 100 - x ) ^{2}} d x

Frequently Asked Questions (FAQ)

What is the integral of sin^6(x/4) ?
The integral of sin^6(x/4) is