How do you find the integral of #f(x) = arctan(x)# ?

1 Answer
CJ
Sep 6, 2014

Integration by parts is a good place to start.

Let #y=uv#. Then #intudv = uv - intvdu# - this is how to integrate by parts.

We need to choose a function u that is easily differentiated, and a function v that is easily antidifferentiated. For this, let's choose #v=x# and #u=arctan(x)#. Hence:
#(dv)/dx = 1 => dv=dx#
#(du)/dx = 1/(1+x^2) => du = dx/(1+x^2)#

Now, we simply substitute these values in to our formula:
#intudv = uv - intvdu#
#therefore#
#intarctan(x)dx = arctan(x)*x - intx*dx/(1+x^2)#

From here, we can antidifferentiate the far right-hand term quite easily, and come up with an answer.

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