What is the integral of sin(x)(1-cos^2(x)) ?

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

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

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

Solution steps

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

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

Add a constant to the solution

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

\int 5 cot^{3} (x) csc^{3} (x) d x

ln ∣ sec (x) ∣

\int sec (5 x) d x

f (x) = 7 e^{x}, at x = e

y^{^{''}} + 10 y^{^{'}} + 29 y = 0

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