What is the integral of 18sin^5(x) ?

The integral of 18sin^5(x) is 18(-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 18sin^5(x)

\int 18 sin^{5} (x) d x

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

Solution steps

\int 18 sin^{5} (x) d x

Take the constant out:

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

= 18 \cdot \int sin^{5} (x) d x

Rewrite using trig identities

= 18 \cdot \int (1 - cos^{2} (x))^{2} sin (x) d x

Apply u-substitution

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

= 18 \cdot \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

= 18 (- \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}

= 18 (- u + \frac{2 u ^{3}}{3} - \frac{u ^{5}}{5})

Substitute back

u = cos (x)

= 18 (- cos (x) + \frac{2 cos ^{3} ( x )}{3} - \frac{cos ^{5} ( x )}{5})

Add a constant to the solution

= 18 (- cos (x) + \frac{2 cos ^{3} ( x )}{3} - \frac{cos ^{5} ( x )}{5}) + C

\int \frac{1}{6 - x ^{2}} d x

\int \frac{3 x + 2}{( 6 x ^{2} + 8 x ) ^{5}} d x

\int + z e^{2 z} d z

\int 9 x^{3} sin (x^{2}) d x

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

What is the integral of 18sin^5(x) ?
The integral of 18sin^5(x) is 18(-cos(x)+(2cos^3(x))/3-(cos^5(x))/5)+C