I Solving a definite integral by differentiation under the integral

[Mr Davis 97]

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?

 

[ShayanJ]

I don't think there is any systematic way for that. Its something you should figure out by yourself whether this method can be used for a particular integral or not. In fact if a particular place comes to your mind that placing the parameter there would let you use this method, then you can consider using this method. Its not the other way around.