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

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

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integral of 2cos^3(x)

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

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

Solution steps

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

Rewrite using trig identities

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

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

Add a constant to the solution

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

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