Answer to Exercise 11.4-2

The problem is to use integration by parts to integrate

arctan(x) dx

Step 1: Choose your parts. When the integrand is an inverse function like arctan, you always choose the parts the way we did in the text when we integrated ln(x).

u = arctan(x)
and
dv = 1 dx

Step 2: What is v? Remember that you find v by integrating the expression you choose for dv/dx. So do that, then click here to see if you got the right expression.

You have dv/dx = 1. So integrating that you get:

v =

dx =

x + C

You remember that in this step you are allowed to drop the C in the above when you stick this in for v into equation 11.4-6a (where you keep 11.4-6a's C). So what you have so far is

x arctan(x) + C -

du x dx = dx

arctan(x) dx

Step 3: What is du/dx? You find that by taking the derivative of u = arctan(x), whose derivative you can find on the table. So do that, then plug the result into the above and click here to move on.

Of course you have

   du        1
       =        
   dx     1 + x²

and when you put that into the integration by parts you have already developed, you get

x arctan(x) + C -

x dx = 1 + x²

arctan(x) dx

Step 4: Do the remaining integral: Well this one is not exactly a piece of cake, but can you see how what you learned in the section on integration by simple substitution applies to this remaining integral.

Step 1a) Look for one part of the integrand that is the derivative of another. There's nothing exactly that matches here, but if you munge it using the little trick you learned in the last section, there is. The equation above is the same as what (that will make the numerator of the integrand equal to the derivative of its denominator)? Figure it out, then click here to move on.

You should have

x arctan(x) + C -

1 2

2x dx = 1 + x²

arctan(x) dx

Step 2a: Preparing the substitution variable. Since we already used the symbol, u, in assigning the parts, we'll use s for making the substitution, which is

   s  =  1 + x²


   ds
       =  2x
   dx


   ds  =  2x dx

So go ahead and make the substitution, then click here.

Step 3a: Making the substitution: Now you have

1 x arctan(x) + C − 2

ds = s

arctan(x) dx

Step 4a: Integrate. You know how to do this one. If not you can look it up on the table. Then click here.

Now you have

1 x arctan(x) - ln|s| + C = 2

arctan(x) dx

Step 5a: Substitute back. Remember that s = 1 + x². Then click here for the final answer.

... which is

1 x arctan(x) - ln|1 + x²| + C = 2

arctan(x) dx

But take its derivative just to be sure.

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