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:
òarctan(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 + x2 and dw = 2x dx. Then
òdx = ò = ò dw = ln|x| + C = ln|1 + x2| + C.
Step 4: Using this result, the answer is:
òarctan(x) dx = x[arctan(x)] - ln|1 + x2| - C.
Note: this result can be written as follows:
òarctan(x) dx = x[arctan(x)] - ln(1 + x2) + C.
Since 1 + x2 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.