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

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

Baholash

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

Yechim qadamlari

Avval noaniq integralni baholang.

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

\int af\left(x\right)\mathrm{d}x=a\int f\left(x\right)\mathrm{d}x yordamidan konstantani qavsdan tashqariga chiqaring.

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

k\neq -1 uchun integral \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} boʻlgani uchun, \int x\mathrm{d}x integralni \frac{x^{2}}{2} bilan almashtiring.

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

Qisqartirish.

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

Xos integral bu integral hisoblashning yuqori chegarasida hisoblangan ifodaning boshlangʻich holatidan chiqarib tashlagan holda integral hisoblashning quyi chegarasida hisoblangan ifodaning boshlangʻich holatidir.

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

Qisqartirish.

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

h ga nisbatan hosilani topish

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