Question

`\int1/(x^2+a^2)dx`?

"Let `a` be a constant, and evaluate the integral."

Answer 1

See process below

Explanation:

`int1/(x^2+a^2)dx`

Lets do the change `x=atantheta`. With this change we have

`dx=asec^2theta d theta`. Put in the integral

`int(cancelasec^2theta d theta)/(a^cancel2(tan^2theta+1))`

But `tan^2theta+1=sec^2theta`. The we have

`int(cancelacancelsec^2theta d theta)/(a^cancel2cancelsec^2theta)=1/aint d theta=1/atheta +C`

Undoing the change `theta= arctan(x/a)`. We have finally

`int1/(x^2+a^2)dx=1/a arctan(x/a)+C`