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I recently discovered Differentiation under the integral sign a.k.a Feynman Integration and I read an article which says it can be substituted for contour integration. Therefore, I am assuming this technique is, indeed, very powerful. I was looking for a list of integrals which are, seemingly, hard but are made easy via this technique.

Thanks a lot!

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closed as too broad by Start wearing purple, David H, Namaste, Najib Idrissi, azimut Jul 26 '14 at 15:39

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ One of them is math.stackexchange.com/questions/874029/… $\endgroup$ – Jack D'Aurizio Jul 25 '14 at 12:26
  • $\begingroup$ Some easier examples of such integrals are $$\int^{\infty}_0\frac{\ln{x}}{1+x^4}dx$$ and $$\int^1_0x^a(\ln{x})^bdx$$ $\endgroup$ – SuperAbound Jul 25 '14 at 12:27
  • $\begingroup$ Wikipedia gives a dozen of examples of different types. You don't find this sufficient? $\endgroup$ – Start wearing purple Jul 25 '14 at 12:28
  • $\begingroup$ @O.L. I thought it would be nice to have a list on math.se with explanations provided by experienced users to look at and furthermore, for people new to the technique to look it's power and see how it applies to a wide range of integrals. $\endgroup$ – Jeel Shah Jul 25 '14 at 13:08
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    $\begingroup$ You should certainly look at math.uconn.edu/~kconrad/blurbs/analysis/diffunderint.pdf for some nice insights by Keith Conrad $\endgroup$ – James S. Cook Jul 25 '14 at 13:20
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Note that $$ \int_0^1 x^\alpha\ dx=\frac1{\alpha+1},\qquad\text{for }\ \alpha>-1.\tag1 $$ Differentiating $(1)$ $n$ times yields $$ \int_0^1 \frac{\partial^n}{\partial\alpha^n}\left(x^\alpha\right)\ dx=\color{blue}{\int_0^1 x^\alpha \ln^n x\ dx=\frac{(-1)^n n!}{(\alpha+1)^{n+1}}}, \qquad\text{for }\ n=0,1,2,\ldots\tag2 $$

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