What is the integral of (sin^3(x)-sin^5(x)) ?

The integral of (sin^3(x)-sin^5(x)) is 1/3 cos^3(x)-2/3 cos^3(x)+1/5 cos^5(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^3(x)-sin^5(x))

\int (sin^{3} (x) - sin^{5} (x)) d x

\frac{1}{3} cos^{3} (x) - \frac{2}{3} cos^{3} (x) + \frac{1}{5} cos^{5} (x) + C

Solution steps

\int sin^{3} (x) - sin^{5} (x) d x

Apply the Sum Rule:

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

= \int sin^{3} (x) d x - \int sin^{5} (x) d x

\int sin^{3} (x) d x = - cos (x) + \frac{cos ^{3} ( x )}{3}

\int sin^{5} (x) d x = - cos (x) + \frac{2 cos ^{3} ( x )}{3} - \frac{cos ^{5} ( x )}{5}

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

Simplify

- cos (x) + \frac{cos ^{3} ( x )}{3} - (- cos (x) + \frac{2 cos ^{3} ( x )}{3} - \frac{cos ^{5} ( x )}{5}) : \frac{1}{3} cos^{3} (x) - \frac{2}{3} cos^{3} (x) + \frac{1}{5} cos^{5} (x)

= \frac{1}{3} cos^{3} (x) - \frac{2}{3} cos^{3} (x) + \frac{1}{5} cos^{5} (x)

Add a constant to the solution

= \frac{1}{3} cos^{3} (x) - \frac{2}{3} cos^{3} (x) + \frac{1}{5} cos^{5} (x) + C

\int cos (2 x) (1 + 4 sin (2 x))^{4} d x

\int \frac{5}{6 x} d x

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

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

\int sec (x) tan (x) + \frac{1}{1 - x ^{2}} d x

What is the integral of (sin^3(x)-sin^5(x)) ?
The integral of (sin^3(x)-sin^5(x)) is 1/3 cos^3(x)-2/3 cos^3(x)+1/5 cos^5(x)+C