Solve ∫ (from 0 to 2 pi) of operatornamelos^2θdθ

$$\int _ { 0 } ^ { 2 \pi } \operatorname { los } ^ { 2 } \theta d \theta$$

Evaluate

$$2\left(\pi los\right)^{2}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \left(los\right)^{2}\theta \mathrm{d}\theta$$

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

$$l^{2}o^{2}s^{2}\int \theta \mathrm{d}\theta$$

Since \int \theta ^{k}\mathrm{d}\theta =\frac{\theta ^{k+1}}{k+1} for k\neq -1, replace \int \theta \mathrm{d}\theta with \frac{\theta ^{2}}{2}.

$$l^{2}o^{2}s^{2}\times \frac{\theta ^{2}}{2}$$

Simplify.

$$\frac{l^{2}o^{2}s^{2}\theta ^{2}}{2}$$

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.

$$\frac{1}{2}l^{2}o^{2}s^{2}\times \left(2\pi \right)^{2}-\frac{1}{2}l^{2}o^{2}s^{2}\times 0^{2}$$

Simplify.

$$2l^{2}o^{2}s^{2}\pi ^{2}$$

Differentiate w.r.t. l

$$4l\left(\pi os\right)^{2}$$