What is the integral of sin^3(x)cos^{20}(x) ?

The integral of sin^3(x)cos^{20}(x) is -(cos^{21}(x))/(21)+(cos^{23}(x))/(23)+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(x)cos^{20}(x)

\int sin^{3} (x) cos^{20} (x) d x

- \frac{cos ^{21} ( x )}{21} + \frac{cos ^{23} ( x )}{23} + C

Solution steps

\int sin^{3} (x) cos^{20} (x) d x

Rewrite using trig identities

= \int (1 - cos^{2} (x)) sin (x) cos^{20} (x) d x

Apply u-substitution

: \int - u^{20} (1 - u^{2}) d u

= \int - u^{20} (1 - u^{2}) d u

Expand

- u^{20} (1 - u^{2}) : - u^{20} + u^{22}

= \int - u^{20} + u^{22} 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^{20} d u + \int u^{22} d u

\int u^{20} d u = \frac{u ^{21}}{21}

\int u^{22} d u = \frac{u ^{23}}{23}

= - \frac{u ^{21}}{21} + \frac{u ^{23}}{23}

Substitute back

u = cos (x)

= - \frac{cos ^{21} ( x )}{21} + \frac{cos ^{23} ( x )}{23}

Add a constant to the solution

= - \frac{cos ^{21} ( x )}{21} + \frac{cos ^{23} ( x )}{23} + C

y^{^{''}} - \frac{2}{7} y^{^{'}} + \frac{1}{49} y = 0

x^{2} \frac{d y}{d x} + 2 x y = 1

x^{3} - 6 x^{2} - 34 x + 40, at x = - 2

\int arcsec (2 x) d x

x \to - 4 lim (x + 3)

What is the integral of sin^3(x)cos^{20}(x) ?
The integral of sin^3(x)cos^{20}(x) is -(cos^{21}(x))/(21)+(cos^{23}(x))/(23)+C