# Integral of x arctan x dx

by alba_ei
Tags: arctan, integral
 P: 38 1. The problem statement, all variables and given/known data $$\int x \arctan x \, dx$$ 3. The attempt at a solution By parts, $$u = \arctan x$$ $$dv = x dx$$ $$du = \frac{dx}{x^2+1}$$ $$v = \frac{x^2}{2}$$ $$\int x \arctan x \, dx = \frac{x^2}{2}\arctan x - \frac{1}{2} \int \frac{x^2}{x^2+1} \, dx$$ Again...by parts $$u = x^2$$ $$dv = \frac{dx}{x^2+1}$$ $$du = 2x dx$$ $$v = arc tan x$$ $$\int x \arctan x \, dx = \frac{x^2}{2}\arctan x - \frac{x^2}{2}\arctan x - \int x \arctan x \, dx$$ I back to the beginning, what did wrogn? $$\int x \arctan x \, dx = - \int x \arctan x \, dx$$
 P: 483 $$\int x \arctan x \, dx = \frac{x^2}{2}\arctan x - \frac{x^2}{2}\arctan x - \int x \arctan x \, dx$$ Add $$\int x \arctan x \, dx$$ to both sides, then solve for the integral, assuming your work is correct.
P: 38
 Quote by z-component $$\int x \arctan x \, dx = \frac{x^2}{2}\arctan x - \frac{x^2}{2}\arctan x - \int x \arctan x \, dx$$ Add $$\int x \arctan x \, dx$$ to both sides, then solve for the integral, assuming your work is correct.
you mean like this? is the same, i back to the beginign

$$\int x \arctan x \, dx +\int x \arctan x \, dx = \frac{x^2}{2}\arctan x - \frac{x^2}{2}\arctan x - \int x \arctan x \, dx +\int x \arctan x \, dx$$

$$2\int x \arctan x \, dx = 0$$

P: 2,046
Integral of x arctan x dx

 Quote by alba_ei $$- \frac{1}{2} \int \frac{x^2}{x^2+1} \, dx$$
Why use 'by parts' again? It would easier if you just add and subtract 1 from the numerator
 P: 49 why not try the substitution u=x^2+1 in that second integral...
 P: 43 for the integral x²/(x²+1) you can rewrite it as (x² + 1 - 1)/(x²+1) => 1 - 1/(x²+1)
 HW Helper P: 3,348 umm hmm, that leaves a nice (x - arctan x) for you there.
 P: 1

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