# Questions tagged [integration]

For questions about the properties of integrals. Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that describe the type of integral being considered. This tag often goes along with the (calculus) tag.

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### Evaluating $\int P(\sin x, \cos x) \text{d}x$

Suppose $\displaystyle P(x,y)$ a polynomial in the variables $x,y$. For example, $\displaystyle x^4$ or $\displaystyle x^3y^2 + 3xy + 1$. Is there a general method which allows us to evaluate the ...
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### Does $\int_0^{\infty}\frac{\sin x}{x}dx$ have an improper Riemann integral or a Lebesgue integral?

In this wikipedia article for improper integrals, $$\int_0^{\infty}\frac{\sin x}{x}dx$$ is given as an example for the integrals that have an improper Riemann integral but do not have a (proper) ...
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### Integrate $\int_0^{\pi/2} \frac{1}{1+\tan^\alpha{x}}\,\mathrm{d}x$

Evaluate the integral $$\int_0^{\pi/2} \frac{1}{1+\tan^\alpha{x}}\,\mathrm{d}x$$
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### Frullani 's theorem in a complex context.

It is possible to prove that $$\int_{0}^{\infty}\frac{e^{-ix}-e^{-x}}{x}dx=-i\frac{\pi}{2}$$ and in this case the Frullani's theorem does not hold since, if we consider the function $f(x)=e^{-x}$, we ...
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### Series as an integral (sophomore's dream)

I need help with this exercise. I need to prove $$\int_{0}^{1}x^{-x}\ dx=\sum_{n=1}^{\infty}n^{-n}$$ I think I should use some convergence theorem, but I'm stuck. Thanks a lot!
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### Can a limit of an integral be moved inside the integral?

After coming across this question: How to verify this limit, I have the following question: When taking the limit of an integral, is it valid to move the limit inside the integral, providing the ...
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