Showing that $\int_0^{\infty} \frac{x^8-4x^6+9x^4-5x^2+1}{x^{12}-10x^{10}+37x^8-42x^6+26x^4-8x^2+1}\,{dx} = \frac{1}{2}\pi$ by real analysis
110602
A nasty integral of a rational function
76408
Show that $\int_0^\pi f(\sin x)\,\mathrm{d}x = 2\int_0^{\pi/2}f(\sin x) \, \mathrm{d}x$
140638
Integrating $\int_0^{\infty} \frac{dx}{1+x^3}$ using residues.
122843
Finding $\int_0^\infty \frac{\cos(ax)-\cos(bx)}{x}dx$
23087
Evaluate $\int_0^\infty \frac{(\log x)^2}{1+x^2} dx$ using complex analysis
24558
Show that $\frac{1}{\sqrt{2\pi}} \cdot \left(\int_{- \infty}^{\infty} e^{itx} \cdot e^{-\frac{1}{2}t^2 } dx \right) = e^{-\frac{1}{2}t^2}$
24022
Compute $\int _0 ^{2\pi} (\sum_{n=1}^\infty \frac {\cos(nx)}{2^n})^2 dx$
30193
Real-Analysis Methods to Evaluate $\int_0^\infty \frac{x^a}{1+x^2}\,dx$, $|a|<1$.
57224
Compute $\int_0^\infty \frac{dx}{1+x^3}$
9390
$ \int_1^2\int_1^2 \int_1^2 \int_1^2 \frac{x_1+x_2+x_3-x_4}{x_1+x_2+x_3+x_4}dx_1dx_2dx_3dx_4 $
143093
Determining k: $\int_{6}^{16} \frac{dx}{\sqrt{x^3 + 7x^2 + 8x - 16}} = \frac{\pi}{k}$
21375
The integral $\int_0^\infty \dfrac{x \sin(x)}{x^2+1} dx$
137060
Evaluating $\int_0^{\tfrac{\pi}{4}}\ln(\cos x-\sin x)\ln(\cos x) dx - \int_0^{\tfrac{\pi}{4}}\ln(\cos x+\sin x)\ln(\sin x)dx=\frac{G\ln 2}{2}$
148743
$\int_0^\infty \frac{1}{1+x^ 9} \, dx$
83703
Show that $\int_{0}^{\infty}|\frac{\sin x}{x}|\,dx=\infty $
68821
Prove $\int_0^{\infty} \left(\sqrt{1+x^{4}}-x^{2}\right)\ dx=\frac{\Gamma^{2}\left(\frac{1}{4}\right)}{6\sqrt{\pi}}$
87242
Ways to prove $ \int_0^\pi dx \frac{\sin^2(n x)}{\sin^2 x} = n\pi$