A few improper integral

calculus improper-integrals integration

Solution 1:

All of them can be solved using the beta function technique. See (I), (II), (III). Notice that, the second integral can be written as,

$$ \int_{-\infty }^{+\infty }{\frac{{{x}^{2m}}}{1+{{x}^{2n}}}\text{d}x}=2\int_{0 }^{+\infty }{\frac{{{x}^{2m}}}{1+{{x}^{2n}}}\text{d}x}. $$

Related

Evaluate $\lim\limits_{n\to\infty}\frac{\sum_{k=1}^n k^m}{n^{m+1}}$
What's the limit of the sequence $\lim\limits_{n \to\infty} \frac{n!}{n^n}$?
Can a countable set contain uncountably many infinite subsets such that the intersection of any two such distinct subsets is finite?
Prove that $\sin(2A)+\sin(2B)+\sin(2C)=4\sin(A)\sin(B)\sin(C)$ when $A,B,C$ are angles of a triangle
About a paper of Zermelo
Why are polynomials defined to be "formal" (vs. functions)?
Largest integer that can't be represented as a non-negative linear combination of $m, n = mn - m - n$? Why?
Prove that if $\gcd( a, b ) = 1$ then $\gcd( ac, b ) = \gcd( c, b ) $
$\sqrt x$ is uniformly continuous
Show that a continuous function has a fixed point
Determinant of a block lower triangular matrix
Subgroups of a cyclic group and their order.