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

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

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

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

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

Solution steps

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

x + 1

\frac{4 s + 7}{s ^{2} - 4}

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

(y + 1) (\frac{d y}{d x}) = 2 x

y^{^{''}} - 7 y^{^{'}} + 12 y = 0, y (0) = - \frac{1}{2}, y^{^{'}} (0) = - \frac{9}{4}

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