Solve ∫ (from 0 to pi) of cos^3θsinphidphi

$$\int _ { 0 } ^ { \pi } \cos ^ { 3 } \theta \sin \phi d \phi$$

Evaluate

$$2\left(\cos(\theta )\right)^{3}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \left(\cos(\theta )\right)^{3}\sin(ϕ)\mathrm{d}ϕ$$

Factor out the constant using \int af\left(ϕ\right)\mathrm{d}ϕ=a\int f\left(ϕ\right)\mathrm{d}ϕ.

$$\left(\cos(\theta )\right)^{3}\int \sin(ϕ)\mathrm{d}ϕ$$

Use \int \sin(\theta )\mathrm{d}\theta =-\cos(\theta ) from the table of common integrals to obtain the result.

$$-\left(\cos(\theta )\right)^{3}\cos(ϕ)$$

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.

$$-\left(\cos(\theta )\right)^{3}\cos(\pi )+\left(\cos(\theta )\right)^{3}\cos(0)$$

Simplify.

$$2\left(\cos(\theta )\right)^{3}$$

Differentiate w.r.t. θ

$$-6\sin(\theta )\left(\cos(\theta )\right)^{2}$$