How do you find the primitive/integral of arctan⁡x?I tried searching for how you derive the integral/primitive of arctan⁡(x), but I can't find any question on S.E with an answer that clearly explains this. I feel that there should be one, since it is rather fundamental knowledge, and it does come up every now and then as part of S.E questions.So the primitive can be looked up and it should be the following:∫arctan⁡(x)dx=xarctan⁡(x)−12ln⁡(1+x2)+CHow do you get the result?

Question

How do you find the primitive/integral of arctan⁡x?I tried searching for how you derive the integral/primitive of arctan⁡(x), but I can&#039;t find any question on S.E with an answer that clearly explains this. I feel that there should be one, since it is rather fundamental knowledge, and it does come up every now and then as part of S.E questions.So the primitive can be looked up and it should be the following:∫arctan⁡(x)dx=xarctan⁡(x)−12ln⁡(1+x2)+CHow do you get the result?

Quimpoah3b · Accepted Answer

The integral can be solved by using integration by parts. The trick is to think of the function as a product of arctan⁡(x) and 1, both with respect to x.We solve the integral by choosing to integrate 1 to x and differentiate arctan⁡(x) into 11+x2 which is easier to integrate:∫arctan⁡(x)⋅1dx=xarctan⁡(x)−∫x1+x2dx+C0and we can use the fact that 12(ddxln⁡(1+x2))=x1+x2 to integrate x1+x2 so the last integration results in the given primitive:xarctan⁡(x)−∫x1+x2∼dx=xarctan⁡(x)−12ln⁡(1+x2)+C

Egerlandsq0z · Accepted Answer

Indeed, if you know a primitive for f, say F, you can derive a primitive for f−1 in the same way: putting x=f(u), so u=f−1(x), we have dx=f′(u)du, so∫f−1(x)dx=∫uf′(u)du=uf(u)−∫f(u)du=f(u)u−F(u)+C=xf−1(x)−F(f−1(x))+CFor example, in this case f(u)=tan⁡u, so a primitive islog|sec⁡{u}|=12log⁡{(1+tan2{u})}=12log⁡{(1+x2)} , and the result follows. One can apply the same rule for log⁡x, and all the other trigonometric and hyperbolic inverses.

How do you find the primitive/integral of

Answered question

Answer & Explanation

New Questions in Integral Calculus