Well I am amazed no one mentioned this yet. But this idea is exactly what makes the Feynman technique of integration effective. You take an integral and you add an additional variable into the expression. This new modified expression can be thought as a family of integral for each value the new variable can take.

You differentiate with the value of the integral w.r.t. the new parameter and do the original integration you had. This would lead you into a differential equation with the new parameter.

You can get the initial value of the DE by choosing a suitable member in the family of integral to evaluate.

Once you have the two informations above you have the parameterized integral as some closed functional form of alpha. Now you choose the value of alpha that returns you to the original integral and youd have solved the question.