What is the integral of sin^5(x) ?

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of sin^5(x)

\int sin^{5} (x) d x

- cos (x) + \frac{2 cos ^{3} ( x )}{3} - \frac{cos ^{5} ( x )}{5} + C

Solution steps

\int sin^{5} (x) d x

Rewrite using trig identities

= \int (1 - cos^{2} (x))^{2} sin (x) d x

Apply u-substitution

: \int - 1 + 2 u^{2} - u^{4} d u

= \int - 1 + 2 u^{2} - u^{4} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - \int 1 d u + \int 2 u^{2} d u - \int u^{4} d u

\int 1 d u = u

\int 2 u^{2} d u = \frac{2 u ^{3}}{3}

\int u^{4} d u = \frac{u ^{5}}{5}

= - u + \frac{2 u ^{3}}{3} - \frac{u ^{5}}{5}

Substitute back

u = cos (x)

= - cos (x) + \frac{2 cos ^{3} ( x )}{3} - \frac{cos ^{5} ( x )}{5}

Add a constant to the solution

= - cos (x) + \frac{2 cos ^{3} ( x )}{3} - \frac{cos ^{5} ( x )}{5} + C

\int sec^{2} (x) d x

\frac{d y}{d x} = \frac{y + 2}{x + 7}

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

y = \frac{ln ( x )}{x ^{5}}

\int \frac{e ^{3 t}}{t} d t

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