What is the integral of 2cos^3(t) ?

The integral of 2cos^3(t) is 2(sin(t)-(sin^3(t))/3)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of 2cos^3(t)

\int 2 cos^{3} (t) d t

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

Solution steps

\int 2 cos^{3} (t) d t

Take the constant out:

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

= 2 \cdot \int cos^{3} (t) d t

Rewrite using trig identities

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

Apply u-substitution

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

= 2 \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

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

\int 1 d u = u

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

= 2 (u - \frac{u ^{3}}{3})

Substitute back

u = sin (t)

= 2 (sin (t) - \frac{sin ^{3} ( t )}{3})

Add a constant to the solution

= 2 (sin (t) - \frac{sin ^{3} ( t )}{3}) + C

t y^{^{'}} - 3 y = t^{4}

\int \frac{x}{3 - x ^{2}} d x

\int \frac{x ^{2} - 2 x + 6}{( x + 1 ) ( x - 2 ) ^{2}} d x

x^{2} (120 - x)

\int 10 sec^{2} (5 x) tan (5 x) d x

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