- #1
Mr Davis 97
- 1,462
- 44
Say we have the following integral: ##\displaystyle \int_0^1 \frac{\log (x+1)}{x^2+1}##. I know how to do this integral with a tangent substitution. However, I saw another method, which was by differentiating ##f## under the integral with respect to the parameter ##t##, where we let ##\displaystyle f(t) = \int_0^1 \frac{\log (tx+1)}{x^2+1}##. This indeed leads to a solution, where after differentiating, we integrate again to get the solution. However, to use this method, first we have to find somewhere to insert the parameter. How does one figure out that the ##t## goes where it does as in this example? If I were trying to do this integral by differentiating under the integral on my own, why would I think to put the ##t## there as opposed to somewhere else?