Problems on Techniques of Integration

Use the Integration by Parts. Set

$$\left\{\begin{array}{lll} u &=& \arctan(x)\\ dv &=& x dx\;. \end{array}\right.$$

Then

$$\left\{\begin{array}{lll} du &=&\displaystyle \frac{1}{x^2 + 1} dx\\ v &=& \displaystyle \frac{x^2}{2}\;. \end{array}\right.$$

So

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

Since

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

we get

$$\int \arctan(x) dx = \frac{x^2}{2} \arctan(x)- \frac{x}{2} + \frac{1}{2}\arctan(x) + C\;.$$

Detailed Answer.

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