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

The integral of-2sin^3(x)cos(x) is -1/2 sin^4(x)+C

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integral of-2sin^3(x)cos(x)

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

- \frac{1}{2} sin^{4} (x) + C

Solution steps

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

Apply u-substitution

: \int u^{3} d u

= - 2 \cdot \int u^{3} d u

Apply the Power Rule

: \frac{u ^{4}}{4}

= - 2 \cdot \frac{u ^{4}}{4}

Substitute back

u = sin (x)

= - 2 \cdot \frac{sin ^{4} ( x )}{4}

Simplify

- 2 \cdot \frac{sin ^{4} ( x )}{4} : - \frac{1}{2} sin^{4} (x)

= - \frac{1}{2} sin^{4} (x)

Add a constant to the solution

= - \frac{1}{2} sin^{4} (x) + C

\int \frac{5 t - 2}{t + 1} d t

\frac{d}{d x} (\frac{x}{12} + 1)

\int_{1}^{2} 2 x^{3} - 4 d x

\frac{\partial}{\partial x} (\frac{5 y ^{2}}{x})

\frac{2 s}{( s ^{2} + 4 ) ^{2}}

What is the integral of-2sin^3(x)cos(x) ?
The integral of-2sin^3(x)cos(x) is -1/2 sin^4(x)+C