Resolver ∫ (from 0 to pi / 3) of (1-cosθ)^2

$$\int _ { 0 } ^ { \pi / 3 } ( 1 - \cos \theta ) ^ { 2 }$$

Calcular

$$\frac{\pi \left(\cos(\theta )-1\right)^{2}}{3}$$

Pasos da solución

Calcula a integral indefinida primeiro.

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

Descubre a integral de \left(1-\cos(\theta )\right)^{2} empregando a táboa de integrais comúns regra \int a\mathrm{d}x=ax.

$$\left(1-\cos(\theta )\right)^{2}x$$

A integral definida é a primitiva da expresión calculada no límite superior de integración menos a primitiva calculada no límite inferior de integración.

$$\frac{1}{3}\left(1-\cos(\theta )\right)^{2}\pi +0\left(1-\cos(\theta )\right)^{2}$$

Simplifica.

$$\frac{\left(\cos(\theta )-1\right)^{2}\pi }{3}$$

Diferenciar w.r.t. θ

$$-\frac{2\pi \sin(\theta )\left(\cos(\theta )-1\right)}{3}$$