Note: In order to do this problem, you need to know that
(arctan(*u*)) =
.

Solution: Step 1: Let *u = *arctan(*x*) and *dv* = 1 *dx*. We find that *du = *
*dx *and *v* = ò *dv* = ò *dx = x*.

Step 2: Substitute these variables into the formula:

ò arctan(*x*)* dx = *ò* u dv = uv* - ò *v* *du = x*[arctan(*x*)]* *- ò
* dx*.

Step 3: Integrate the integral by using the Method of Substitution. Let *w* = 1 + *x*^{2} and *dw* = 2*x* *dx*. Then

ò
* dx = *ò
=
ò
*dw* =
ln|*x*| + *C* =
ln|1 + *x*^{2}| + *C*.

Step 4: Using this result, the answer is:

ò arctan(*x*)* dx* = *x*[arctan(*x*)] -
ln|1 + *x*^{2}| - *C*.

Note: this result can be written as follows:

ò arctan(*x*)* dx* = *x*[arctan(*x*)] -
ln(1 + *x*^{2}) + *C*.

Since 1 + *x*^{2} is postive for all real values of *x*, the absolute value signs can be dropped. Also, subtracting an arbitrary constant is the same as adding one.

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