What is the integral of (sin(x))^8(cos(x))^3 ?

The integral of (sin(x))^8(cos(x))^3 is (sin^9(x))/9-(sin^{11}(x))/(11)+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))^8(cos(x))^3

\int (sin (x))^{8} (cos (x))^{3} d x

\frac{sin ^{9} ( x )}{9} - \frac{sin ^{11} ( x )}{11} + C

Solution steps

\int (sin (x))^{8} (cos (x))^{3} d x

Simplify

= \int sin^{8} (x) cos^{3} (x) d x

Rewrite using trig identities

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

Apply u-substitution

: \int u^{8} (1 - u^{2}) d u

= \int u^{8} (1 - u^{2}) d u

Expand

u^{8} (1 - u^{2}) : u^{8} - u^{10}

= \int u^{8} - u^{10} 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 u^{8} d u - \int u^{10} d u

\int u^{8} d u = \frac{u ^{9}}{9}

\int u^{10} d u = \frac{u ^{11}}{11}

= \frac{u ^{9}}{9} - \frac{u ^{11}}{11}

Substitute back

u = sin (x)

= \frac{sin ^{9} ( x )}{9} - \frac{sin ^{11} ( x )}{11}

Add a constant to the solution

= \frac{sin ^{9} ( x )}{9} - \frac{sin ^{11} ( x )}{11} + C

\int \frac{1}{1 + 5 x} d x

\int 200 + 2 x - 0.003 x^{2} d x

\int \frac{3 e ^{2 x}}{e ^{x} + 1} d x

\int \frac{1}{x ^{2} - \frac{1}{4}} d x

\int \frac{1}{A y + 2} d y

What is the integral of (sin(x))^8(cos(x))^3 ?
The integral of (sin(x))^8(cos(x))^3 is (sin^9(x))/9-(sin^{11}(x))/(11)+C