Solve ∫ (from 0 to frac { pi) of {2}}-sin^2θcosθ

$$\int _ { 0 } ^ { \frac { \pi } { 2 } } - \sin ^ { 2 } \theta \cos \theta$$

Evaluate

$$-\frac{\pi \sin(\theta )\sin(2\theta )}{4}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int -\left(\sin(\theta )\right)^{2}\cos(\theta )\mathrm{d}x$$

Find the integral of -\left(\sin(\theta )\right)^{2}\cos(\theta ) using the table of common integrals rule \int a\mathrm{d}x=ax.

$$-\left(\sin(\theta )\right)^{2}\cos(\theta )x$$

The definite integral is the antiderivative of the expression evaluated at the upper limit of integration minus the antiderivative evaluated at the lower limit of integration.

$$-\frac{1}{2}\left(\sin(\theta )\right)^{2}\cos(\theta )\pi +0\left(\sin(\theta )\right)^{2}\cos(\theta )$$

Simplify.

$$-\frac{\pi \sin(2\theta )\sin(\theta )}{4}$$

Differentiate w.r.t. θ

$$\frac{\pi \left(-\sin(2\theta )\cos(\theta )-2\sin(\theta )\cos(2\theta )\right)}{4}$$