I was recently looking through integration techniques when I came upon differentiation under the integral sign (DUIS). It seems to be pretty powerful, for example:
$$f(t)=\int_0^1\frac{x^t-1}{\ln(x)}\ dx\implies f'(t)=\int_0^1x^t\ dx=\frac1{t+1}\\\implies f(t)=C+\ln(t+1)\\f(0)=0\implies C=0\\\implies\int_0^1\frac{x^t-1}{\ln(x)}\ dx=\ln(t+1)$$
Now, on my own, proving that $\int_0^1\frac{x^t-1}{\ln(x)}\ dx=\ln(t+1)$ would've been a hefty task without DUIS, and so I wanted to ask this question:
It's hard to produce interesting examples where applying DUIS is almost magical, so what are some good example uses of DUIS?
Here is a rare example involving high-order DUIS
$$I_n(t)=\int_{0}^{2\pi}e^{t\cos\theta}\cos(n\theta+t\sin\theta)d\theta $$ Differentiate to establish $$I_n’(t)= I_{n+1}(t)$$ along with $I_1(t)=0$. which results in \begin{align} I_{n}(t) = \frac{d^{n-1}}{dt^{n-1}}I_1(t)=0 \end{align}