Differentiation under integration sign technique

Question

What is Richard Feynman's 'differentiation under integration sign' technique?

Please state the technique and give an example.

I am a Physics undergrad.

Answer 1

Suppose, you want to calculate $\int x\cdot e^x dx$

Now start with $$F(a)=\int e^{ax}dx=\frac{e^{ax}}{a}+C$$

Differntiating with respect to $a$ gives $$F'(a)=\int xe^{ax}dx=\frac{xe^{ax}\cdot a-e^{ax}}{a^2}$$ due to the quotient rule

For $a=1$, we get $$\int x\cdot e^xdx=xe^x-e^x$$