# I tried to solve the integral $\int\frac{\arctan{x}}{1+x^2}dx$ and I got lost. I have the answer, but I have no idea how to get there.

I do not know how to integrate

$$\int{\arctan{x} \over 1+x^2}dx$$

The answer is given to be: $\arctan{x} + 1/2 \ln{|1+x^2|}+C$

I'd appreciate your help very much. Thanks!

Edit 1: it must be done with substitution only.

Edit 2: Yes. It's not the answer. Thank you all. I'll delete this in a minute. Thanks!!

• If I am reading the question right, I have trouble with the given answer. Let $u=\arctan x$. Sep 17, 2015 at 5:27
• If the integrand is correct, the given answer is wrong ! Sep 17, 2015 at 5:31
• Hi and welcome to Math.SE. Please share your thoughts on the problem. The answer given is clearly wrong (just differentiate it, and you'll see). Is the problem written correctly as it is now, when edited? Also, in the future, you could write your questions using mathjax. Sep 17, 2015 at 5:33
• Could it be? My teacher gave me that answer. Maybe you're right, he's wrong. Sep 17, 2015 at 5:33
• This is a related question, but about a definite integral: What is the value of $\int_0^1 \frac{\arctan x}{1+x^{2}} dx$? Sep 17, 2015 at 16:11

If you set $u=\arctan{x}$ you get $du={dx\over 1+x^2}$ and the integral rewrites as

$$\int udu={u^2\over 2}+C$$

And I share the trouble of Andre Nicolas and Claude Leibovici and it has nothing to do with the fact that the three of us are French

If you derive the "answer" that's given you get

$${1+x\over 1+x^2}\neq {\arctan{x}\over 1+x^2}$$

• Perhaps French is forever, but was French is more accurate. Sep 17, 2015 at 5:54
• Could anyone suspect a French coalition on this site ? Cheers :-) Sep 17, 2015 at 5:54

Notice that $ff'$ is the derivative of $\frac12 f^2+C$ and in your example $f(x)=\arctan x$.