# Integration by Parts: Int: x*arctan(x) dx

Because of circumstance (my desire to graduate in 5 years or less), I've been forced to attempt Calc 2 in 2 months time online over the summer. About 75% of it is going smoothly (compared with 105% or so of Calc 1).

## Homework Statement

I'm to solve the indefinite integral: $$\int$$ x * arctan(x) dx

## Homework Equations

Integration by parts is done using: $$\int$$u dv = uv - $$\int$$v du

## The Attempt at a Solution

It seems pretty obvious that u = arctan(x) and that dv = x. From this, du = $$\frac{1}{1+x^2}$$ dx and v = $$\frac{1}{2}$$x2.

Using the integration by parts formula:

$$\int$$ x * arctan(x) dx = $$\frac{1}{2}$$x2arctan(x) - $$\frac{1}{2}$$$$\int$$$$\frac{x^2}{1 + x^2}$$dx

Now integration by parts must be used again. It seems obvious to select u = x2, dv = $$\frac{1}{1 + x^2}$$. du = 2xdx, dv = arctan(x).

$$\int$$ x * arctan(x) dx = $$\frac{1}{2}$$x2arctan(x) - $$\frac{1}{2}$$ ( x2arctan(x) - 2 $$\int$$xarctan(x) ).

Simplified:

$$\int$$ x * arctan(x) dx = 0 + $$\int$$ x * arctan(x) dx.

While this is very true, it doesn't help me find the integral. Switching my u and dv in either use of the integration by parts formula hasn't yielded a solution for me in my attempts yet. Thank you in advance for your help .

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rock.freak667
Homework Helper
$$\frac{x^2}{x^2+1}=\frac{x^2+1 -1}{x^2+1}=1-\frac{1}{x^2+1}$$

You are my Bokonon, only you tell truths.

Thank you, in other words.