Solve ∫ (from 0 to pi) of sqrt{2+2cosθ}

$$\int _ { 0 } ^ { \pi } \sqrt { 2 + 2 \cos \theta }$$

Evaluate

$$\pi \sqrt{2\left(\cos(\theta )+1\right)}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \sqrt{2\left(1+\cos(\theta )\right)}\mathrm{d}x$$

Find the integral of \sqrt{2\left(1+\cos(\theta )\right)} using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\sqrt{2\left(1+\cos(\theta )\right)}x$$

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.

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

Simplify.

$$\pi \sqrt{2\left(1+\cos(\theta )\right)}$$

Differentiate w.r.t. θ

$$-\frac{\pi \sqrt{\frac{2}{\cos(\theta )+1}}\sin(\theta )}{2}$$