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

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

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integral of 2sin^2(x)cos(x)

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

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

Solution steps

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

Apply u-substitution

: \int u^{2} d u

= 2 \cdot \int u^{2} d u

Apply the Power Rule

: \frac{u ^{3}}{3}

= 2 \cdot \frac{u ^{3}}{3}

Substitute back

u = sin (x)

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

Simplify

2 \cdot \frac{sin ^{3} ( x )}{3} : \frac{2}{3} sin^{3} (x)

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

Add a constant to the solution

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

\int 2 x x + 3 d x

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\int 64 x^{3} (4 x^{2} + 1)^{3} d x

\int x^{2} (x^{3} + 2)^{5} d x

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