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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?

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    Feynman was a big fan of differentiating under the integral. These notes from Keith Conrad look interesting: http://www.math.uconn.edu/~kconrad/blurbs/analysis/diffunderint.pdf Also these notes might be interesting: https://web.williams.edu/Mathematics/lg5/Feynman.pdf See also: "Integration: The Feynman Way": http://fy.chalmers.se/~tfkhj/FeynmanIntegration.pdf – 2017-02-23
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    @littleO Ah thanks! It even has my example on page nine. :D – 2017-02-23
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    Wikipedia has a good list [here](https://en.wikipedia.org/wiki/Leibniz_integral_rule#Examples_for_evaluating_a_definite_integral), 8 examples including your own – 2017-02-23
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    This is really the sort of thing that you can just Google. – 2017-02-23
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    @MathematicsStudent1122 well yeah, but some friends and I wanted a list of interesting integrals here on MSE. – 2017-02-23

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