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

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

Evalwa

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

Passi tas-Soluzzjonijiet

L-ewwel evalwa l-integru indefinit.

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

Sib l-integru ta' \left(1-\cos(\theta )\right)^{2} billi tuża t-tabella tar-regola tal-integrali komuni \int a\mathrm{d}x=ax.

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

L-integru definit huwa l-antiderivattiv tal-espressjoni evalwata fil-limitu superjuri tal-integrazzjoni minus l-antiderivattiv evalwat fil-limitu inferjuri tal-integrazzjoni.

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

Issimplifika.

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

Iddifferenzja w.r.t. θ

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