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

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

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

- \frac{cos ^{9} ( v )}{9} + \frac{cos ^{11} ( v )}{11} + C

Solution steps

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

Rewrite using trig identities

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

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 = cos (v)

= - \frac{cos ^{9} ( v )}{9} + \frac{cos ^{11} ( v )}{11}

Add a constant to the solution

= - \frac{cos ^{9} ( v )}{9} + \frac{cos ^{11} ( v )}{11} + C

\frac{d}{d n} ((n + 1)^{n})

f (x) = \frac{sin ( 5 x )}{3 x}

\frac{7}{1 - x}

\frac{d}{d x} (\frac{12}{( 1 - x ) ^{5}})

n = 3 \sum \infty \frac{x ^{n}}{n !}

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