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

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

Evaluate

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

Solution Steps

Evaluate the indefinite integral first.

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

Factor out the constant using \int af\left(x\right)\mathrm{d}x=a\int f\left(x\right)\mathrm{d}x.

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

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x\mathrm{d}x with \frac{x^{2}}{2}.

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

Simplify.

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

The definite integral is the antiderivative of the expression evaluated at the upper limit of integration minus the antiderivative evaluated at the lower limit of integration.

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

Simplify.

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

Differentiate w.r.t. h

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