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

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

Enrhifo

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

Camau Ateb

Gwerthuso’r integryn amhenodol yn gyntaf.

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

Ffactoreiddio allan y cysonyn gan ddefnyddio \int af\left(x\right)\mathrm{d}x=a\int f\left(x\right)\mathrm{d}x.

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

Ers \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} ar gyfer k\neq -1, disodli \int x\mathrm{d}x gyda \frac{x^{2}}{2}.

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

Symleiddio.

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

Yr integryn pendant yw integryn amhendant y mynegiant wedi’i werthuso ar lefel uchaf yr integreiddiad llai’r integryn amhendant ar lefel isaf yr integreiddiad.

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

Symleiddio.

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

Gwahaniaethu w.r.t. h

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