Solve ∫ (from 0 to pi / 2) of sin^9θcos^5θ

$$\int _ { 0 } ^ { \pi / 2 } \sin ^ { 9 } \theta \cos ^ { 5 } \theta$$

Evaluate

$$\frac{\pi \left(\cos(\theta )\right)^{5}\left(\sin(\theta )\right)^{9}}{2}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \left(\sin(\theta )\right)^{9}\left(\cos(\theta )\right)^{5}\mathrm{d}x$$

Find the integral of \left(\sin(\theta )\right)^{9}\left(\cos(\theta )\right)^{5} using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\left(\sin(\theta )\right)^{9}\left(\cos(\theta )\right)^{5}x$$

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.

$$\frac{1}{2}\left(\sin(\theta )\right)^{9}\left(\cos(\theta )\right)^{5}\pi +0\left(\sin(\theta )\right)^{9}\left(\cos(\theta )\right)^{5}$$

Simplify.

$$\frac{\left(\sin(\theta )\right)^{9}\left(\cos(\theta )\right)^{5}\pi }{2}$$

Differentiate w.r.t. θ

$$\frac{\pi \left(9\left(\cos(\theta )\right)^{2}-5\left(\sin(\theta )\right)^{2}\right)\left(\cos(\theta )\right)^{4}\left(\sin(\theta )\right)^{8}}{2}$$