What is the integral of 2sin^3(x) ?

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

integral of 2sin^3(x)

\int 2 sin^{3} (x) d x

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

Solution steps

\int 2 sin^{3} (x) d x

Take the constant out:

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

= 2 \cdot \int sin^{3} (x) d x

Rewrite using trig identities

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

Apply u-substitution

= 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 = cos (x)

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

Add a constant to the solution

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

x \to \infty lim (x a)

\frac{d}{d x} (\frac{x ^{3} - 6}{x ^{2}})

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

\frac{d}{d x} (arctan (sin (ln (x^{3}))))

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

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