What is the integral of sin^{99}(x)cos^3(x) ?

The integral of sin^{99}(x)cos^3(x) is 1/2 (1/50 sin^{100}(x)-1/51 sin^{102}(x))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of sin^{99}(x)cos^3(x)

\int sin^{99} (x) cos^{3} (x) d x

\frac{1}{2} (\frac{1}{50} sin^{100} (x) - \frac{1}{51} sin^{102} (x)) + C

Solution steps

\int sin^{99} (x) cos^{3} (x) d x

Rewrite using trig identities

= \int (1 - cos^{2} (x))^{49} sin (x) cos^{3} (x) d x

Apply u-substitution

: \int - u^{3} (1 - u^{2})^{49} d u

= \int - u^{3} (1 - u^{2})^{49} d u

Take the constant out:

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

= - \int u^{3} (1 - u^{2})^{49} d u

Apply u-substitution

: \int - \frac{( - v + 1 ) v ^{49}}{2} d v

= - \int - \frac{( - v + 1 ) v ^{49}}{2} d v

Take the constant out:

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

= - (- \frac{1}{2} \cdot \int (- v + 1) v^{49} d v)

Expand

(- v + 1) v^{49} : - v^{50} + v^{49}

= - (- \frac{1}{2} \cdot \int - v^{50} + v^{49} d v)

Apply the Sum Rule:

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

= - (- \frac{1}{2} (- \int v^{50} d v + \int v^{49} d v))

\int v^{50} d v = \frac{v ^{51}}{51}

\int v^{49} d v = \frac{v ^{50}}{50}

= - (- \frac{1}{2} (- \frac{v ^{51}}{51} + \frac{v ^{50}}{50}))

Substitute back

= - (- \frac{1}{2} (- \frac{( 1 - cos ^{2} ( x ) ) ^{51}}{51} + \frac{( 1 - cos ^{2} ( x ) ) ^{50}}{50}))

Simplify

- (- \frac{1}{2} (- \frac{( 1 - cos ^{2} ( x ) ) ^{51}}{51} + \frac{( 1 - cos ^{2} ( x ) ) ^{50}}{50})) : \frac{1}{2} (\frac{1}{50} sin^{100} (x) - \frac{1}{51} sin^{102} (x))

= \frac{1}{2} (\frac{1}{50} sin^{100} (x) - \frac{1}{51} sin^{102} (x))

Add a constant to the solution

= \frac{1}{2} (\frac{1}{50} sin^{100} (x) - \frac{1}{51} sin^{102} (x)) + C

\int \frac{4 ^{x} + 5 \cdot 1 6 ^{x}}{1 + 1 6 ^{x}} d x

\int x^{8} (1 + x^{3})^{\frac{1}{2}} d x

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

\int (1 - cot (5 x))^{2} d x

\int \frac{x - 1}{1 + x ^{3}} d x

What is the integral of sin^{99}(x)cos^3(x) ?
The integral of sin^{99}(x)cos^3(x) is 1/2 (1/50 sin^{100}(x)-1/51 sin^{102}(x))+C