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Is there a rapider or more elegant way to evaluate $\int_0^{+\infty} \frac{\cos(\pi x)\ \text{d}x}{e^{2\pi \sqrt{x}}-1}$?

Inequality $\int^{1}_{0}(u(x))^2\,\mathrm{d}x \leq \frac{1}{6}\int^{1}_{0} (u'(x))^2\,\mathrm{d}x+\left(\int_{0}^{1} u(x)\,\mathrm{d}x \right)^2$

Calculating $\int_0^1 \frac{\operatorname{arctanh}\left(\sqrt{1-\frac{u}{2}}\right)\sqrt{\frac{2 \pi \sqrt{1-u}}{u-2}+\pi } }{u\sqrt{1-u}} \, du$

Compute: $\lim\limits_{n\to+\infty}\int\limits_{0}^1 e^{\{nx\}}x^{100}dx$

A twisted hypergeometric series $\sum_{n=1}^\infty\frac{H_n}{n}\left(\frac{(2n)!}{4^n(n!)^2}\right)^2$

Example for non-Riemann integrable functions

How to evaluate $\int_0^1 {\cos(tx)\over \sqrt{1+x^2}}\,\mathrm{d}x$?

If $f(x)=\int_{x-1}^x f(s)ds$, is $f$ constant? Periodic?

Generalized logarithm integral $\int^1_0 \frac{\log(1+x)}{1+a^2x^2}\, dx$

Find $\int_0^1\frac{\ln^2(1-x)}{x}\ dx$

A Nasty Elliptic Integral

Problem with calculation this integral: $\int_0^\pi \frac{dx}{1+3\sin^2x}$

Challenging problem: Find $a$ where $\int_0^\infty \frac{\cos(ax)\ln(1+x^2)}{\sqrt{1+x^2}}dx=0$.

How to find $\int_0^{\infty}\frac{dx}{(1+x^2)^4}$

Volume of a pyramid, using an integral

Integrating with respect to different variables

Integral $\int_0^{\frac{\pi}{2}} x^2 \sqrt{\sin x} \, dx$

New posts in integration

Is there a rapider or more elegant way to evaluate $\int_0^{+\infty} \frac{\cos(\pi x)\ \text{d}x}{e^{2\pi \sqrt{x}}-1}$?

Reduction formula for $I_{n}=\int {\cos{nx} \over \cos{x}}\rm{d}x$

$\int dx/(x^{10000}-1)$

Calculus Question: Improper integral $\int_{0}^{\infty}\frac{\cos(2x+1)}{\sqrt[3]{x}}\text dx$

Inequality $\int^{1}_{0}(u(x))^2\,\mathrm{d}x \leq \frac{1}{6}\int^{1}_{0} (u'(x))^2\,\mathrm{d}x+\left(\int_{0}^{1} u(x)\,\mathrm{d}x \right)^2$

Calculating $\int_0^1 \frac{\operatorname{arctanh}\left(\sqrt{1-\frac{u}{2}}\right)\sqrt{\frac{2 \pi \sqrt{1-u}}{u-2}+\pi } }{u\sqrt{1-u}} \, du$

Compute: $\lim\limits_{n\to+\infty}\int\limits_{0}^1 e^{\{nx\}}x^{100}dx$

A twisted hypergeometric series $\sum_{n=1}^\infty\frac{H_n}{n}\left(\frac{(2n)!}{4^n(n!)^2}\right)^2$

Example for non-Riemann integrable functions

How to evaluate $\int_0^1 {\cos(tx)\over \sqrt{1+x^2}}\,\mathrm{d}x$?

If $f(x)=\int_{x-1}^x f(s)ds$, is $f$ constant? Periodic?

Generalized logarithm integral $\int^1_0 \frac{\log(1+x)}{1+a^2x^2}\, dx$

Find $\int_0^1\frac{\ln^2(1-x)}{x}\ dx$

A Nasty Elliptic Integral

Problem with calculation this integral: $\int_0^\pi \frac{dx}{1+3\sin^2x}$

Challenging problem: Find $a$ where $\int_0^\infty \frac{\cos(ax)\ln(1+x^2)}{\sqrt{1+x^2}}dx=0$.

How to find $\int_0^{\infty}\frac{dx}{(1+x^2)^4}$

Volume of a pyramid, using an integral

Integrating with respect to different variables

Integral $\int_0^{\frac{\pi}{2}} x^2 \sqrt{\sin x} \, dx$