Issolvi ∫ (from 0 to 2 pi) of operatorname{arctanh}(x) wrt x

$$\int _ { 0 } ^ { 2 \pi } \operatorname { arctanh } ( x ) d x$$

Evalwa

$$2\pi ^{2}\arctan(h)$$

Passi tas-Soluzzjonijiet

L-ewwel evalwa l-integru indefinit.

$$\int \arctan(h)x\mathrm{d}x$$

Iffattura ‘l barra l-kostanti bl-użu ta' \int af\left(x\right)\mathrm{d}x=a\int f\left(x\right)\mathrm{d}x.

$$\arctan(h)\int x\mathrm{d}x$$

Minn \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} għal k\neq -1, issostitwixxi \int x\mathrm{d}x ma' \frac{x^{2}}{2}.

$$\arctan(h)\times \frac{x^{2}}{2}$$

Issimplifika.

$$\frac{\arctan(h)x^{2}}{2}$$

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

$$\frac{1}{2}\arctan(h)\times \left(2\pi \right)^{2}-\frac{1}{2}\arctan(h)\times 0^{2}$$

Issimplifika.

$$2\arctan(h)\pi ^{2}$$

Iddifferenzja w.r.t. h

$$\frac{2\pi ^{2}}{h^{2}+1}$$