en

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of cos^3(x)

Solution

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

Solution

sin (x) - \frac{sin ^{3} ( x )}{3} + C

Solution steps

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

Rewrite using trig identities

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

Apply u-substitution

: \int 1 - u^{2} d u

= \int 1 - u^{2} 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 1 d u - \int u^{2} d u

\int 1 d u = u

\int u^{2} d u = \frac{u ^{3}}{3}

= u - \frac{u ^{3}}{3}

Substitute back

u = sin (x)

= sin (x) - \frac{sin ^{3} ( x )}{3}

Add a constant to the solution

= sin (x) - \frac{sin ^{3} ( x )}{3} + C

Popular Examples

integral of (x^3+3x)e^{-6x}

\int (x^{3} + 3 x) e^{- 6 x} d x

(\partial)/(\partial y)(xe^{x^2-y^2})

\frac{\partial}{\partial y} (x e^{x^{2} - y^{2}})

integral of (4x+3)^9

\int (4 x + 3)^{9} d x

derivative of y=1n(1n(x^3))derivative of

y = 1 n (1 n (x^{3}))

integral of 7sin^4(x)

\int 7 sin^{4} (x) d x

Frequently Asked Questions (FAQ)

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