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There are some definite integrals whose indefinite form can not be evaluated. For example the Gaussian integral, the integral of sinx/x etc. For the first we could use double integration and the apply the polar substitutions while for the second one may use Laplace transformations. Could you please give a list of the most important definite integrals whose indefinite form can not be solved? And also what are the most interesting and important techniques to evaluate such definite integrals? Are there good references including methods to solve them?

Any help will be appreciated.

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One example is $\int \frac {\cos (x)}{1+e^x} $ you cant evaluate as it does not have elementary integral. I dont know much about complex calculus. But if you have something like $I=\int _{-a} ^a \frac {\cos (x)}{1+e^x} $ then we use the property that integral of $f (x)=f (a+b-x) $ so we now have to calculate $I=\int _{-a} ^a \frac {e^x\cos (x)}{1+e^x} $ thus adding previous and new one we have $2I=\int _{-a} ^a \cos (x) $ which is pretty simple. Another example can be $\int \frac {1}{1+\tan^a (x)} $ where $a \in R $ . We cant compute it but we can easily compute $\int _0 ^{\frac {\pi}{2}} \frac {1}{1+\tan^a (x)} $ using same trick as above.

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