What is the integral of 1 (cos^3(x)) ?

The integral of 1 (cos^3(x)) is d(sin(x)-(sin^3(x))/3)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of 1 (cos^3(x))

\int \frac{d}{1} (cos^{3} (x)) d x

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

Solution steps

\int \frac{d}{1} cos^{3} (x) d x

Take the constant out:

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

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

Rewrite using trig identities

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

Apply u-substitution

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

= d \cdot \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

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

\int 1 d u = u

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

= d (u - \frac{u ^{3}}{3})

Substitute back

u = sin (x)

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

Add a constant to the solution

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

\int x^{3} 7^{x^{4}} d x

\int \frac{1}{2 x ^{2} + 9} d x

\int \frac{cos ( \frac{π}{x ^{39}} )}{x ^{40}} d x

\int \frac{e ^{5 x + 2}}{5 x + 2} d x

\int y y + 9 d y

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