Any clue regarding this:  $$\int_0^\infty e^{-a^2x^2} \cos (bx)  dx = \frac {\sqrt \pi}{2a}e^{\frac{-b^2}{4a^2}}$$
1230
Prove that :  $$ \frac{\sqrt{\pi}}{2} e^{-\frac{a^2}{4} } =\int_0^{\infty} e^{-x^2} \cos( a x) \ \mathrm{d}x$$    the only thing I can think of is differentiating the RHS and trying to get :  $$ -2 f'(a) =a f(a) $$ But I couldn't do it.  Can anyone show me how to do this ?
43297
Can anyone give me a hint to evaluate this integral?  $$\int_0^\infty \frac{dx}{1+x^4}$$  I know it will involve the gamma function, but how?
107906
I have to verify that     $$\int_0^\pi \ln(1+\alpha\cos(x))dx=\pi\ln\left(\frac{1+\sqrt{1-\alpha^2}}{2}\right)$$   with $|\alpha|&lt;1$. It is my homework and don't know where to begin.
62670
Does anyone have  a proof for $$\int_0^{\infty}\frac{\sin(x^2)}{x^2}\,dx=\sqrt{\frac{\pi}{2}}.$$ I tried to get it from contour integrating $$\frac{e^{iz^2}-1}{z^2},$$ but failed. Thanks.
60481
I encountered a strange integral with a strange result.  $$\int_0^{\frac{\pi}{2}}\log \left( x^2+\log^2(\cos x)\right)dx = \pi \log \left(\log (2) \right)$$  Believe it or not, the result agrees numerically.   How can we prove this result?   Please help me. I feel very curious to know how this can proved.
109003
What is the value of $$\int_0^{\pi} e^{a\cos(t)}\cos(a\sin t)dt?$$
105134
Need help with this integration:   Let $$A = \int_0^\pi \frac{\cos x}{(x+2)^2}dx$$ Compute $$\int_0^{\frac{\pi}{2}} \frac{\sin x \cos x}{x+1}dx$$  In terms of $A$.  I tried to do some algebraic manipulations but either get something rather complicated or just dead ends. Any help would be great!
84841
There was a friend asking me how to prove  $$\int_0^\infty x^{-x}\,dx&lt;2$$  Mathematica showed that its approximate value is 1.99546, so I think it isn't easy to solve it, can you provide me some ideas about this question?
4558
Show that $$\int\limits_0^\pi{\frac{{{x^2}}}{{\sqrt 5-2\cos x}}}\operatorname d\!x =\frac{{{\pi^3}}}{{15}}+2\pi \ln^2 \left({\frac{{1+\sqrt 5 }}{2}}\right).$$   I don't have any idea how to start, but .
79510
I think this question would be with complicated methods in singularity or other kind of methods. I need to solve the integral $$\int_0^\infty\frac{\cos(ax)-\cos(bx)}{x}dx$$ Thanks a lot.
72470
I am trying to solve this integral, I think that it could be solve using the complex.   $$ \int \limits _0 ^{2\pi} \dfrac{dx}{(\alpha +\beta\cos x)^2} $$
50576
I want to show that $$ \int_{0}^{\infty} J_{n}(bx) e^{-ax} \, dx = \frac{(\sqrt{a^{2}+b^{2}}-a)^{n}}{b^{n}\sqrt{a^{2}+b^{2}}}\ , \quad \ (n \in \mathbb{Z}_{\ge 0} \, , \text{Re}(a) &gt;0 , \,  b &gt;0 ),$$ where $J_{n}(x)$ is the Bessel function of the first kind of order $n$.  But the result I get using an integral representation of $J_{n}(bx)$ is off by a factor of $ \displaystyle \frac{1}{b}$, and I don't understand why.  $$ \begin{align} \int_{0}^{\infty} J_{n}(bx) e^{-ax} \, dx &amp;=  \frac{1}{2 \pi} \int_{0}^{\infty} \int_{-\pi}^{\pi}  e^{i(n \theta -bx \sin \theta)} e^{-ax} \, d \theta \, dx \\ &amp;=  \frac{1}{2 \pi} \int_{-\pi}^{\pi} \int_{0}^{\infty} e^{i n \theta} e^{-(a+ib \sin \theta)x} \, dx \, d \theta \\ &amp;=  \frac{1}{2 \pi} \int_{-\pi}^{\pi} \frac{e^{i n \theta}}{a + ib \sin \theta} \, d \theta  \\  &amp;= \frac{1}{2 \pi} \int_{|z|=1} \frac{z^{n}}{a+\frac{b}{2} \left(z-\frac{1}{z} \right)} \frac{dz} {iz} \\ &amp;= \frac{1}{i\pi} \int_{|z|=1} \frac{z^{n}}{bz^{2}+2az-b} \, dz  \end{align}$$  The integrand has simple poles at $\displaystyle z= -\frac{a}{b} \pm \frac{\sqrt{a^{2}+b^{2}}}{b}$.   But only the pole at $\displaystyle z= -\frac{a}{b} + \frac{\sqrt{a^{2}+b^{2}}}{b}$ is inside the unit circle.  Therefore,  $$ \begin{align} \int_{0}^{\infty} J_{n}(bx) e^{-ax} \, dx &amp;= \frac{1}{i \pi} \, 2 \pi i \ \text{Res} \left[ \frac{z^{n}}{bz^{2}+2az-b}, -\frac{a}{b} + \frac{\sqrt{a^{2}+b^{2}}}{b} \right] \\ &amp;= {\color{red}{b}} \  \frac{(\sqrt{a^{2}+b^{2}}-a)^{n}}{b^{n}\sqrt{a^{2}+b^{2}}} . \end{align}$$
107759
Compute $$ \int_{0}^{ \infty }\frac{\ln xdx}{x^{2}+ax+b}$$ I tried to compute the indefinite integral by factorisation and partial fraction decomposition but it became nasty pretty soon. There must be another way to directly evaluate it without actually computing the indefinite integral which I don't know!
43485
Calculate the integral$$ \int_0^\infty \frac{x \sin rx }{a^2+x^2} dx=\frac{1}{2}\int_{-\infty}^\infty \frac{x \sin rx }{a^2+x^2} dx,\quad a,r \in \mathbb{R}. $$  Edit: I was able to solve the integral using complex analysis, and now I want to try and solve it using only real analysis techniques.
111619
I need help evaluating the following integral $$\int_{0}^{\infty} \frac{\sin^4 x}{x^2}\,dx$$ which should probably be equal to $\frac{\pi}{4}$  Using some trigonometric manipulations I got $\frac{3}{8} - \frac{\cos{2x}}{2} + \frac{\cos{4x}}{8}$ which using integration by parts doesn't lead me to anything pretty.  Update: Not sure if I should post this as a separate question but getting explanation why $\int_{0}^{\infty} \frac{\sin{ax}}{x} = \frac{\pi}{2}$ for a positive integer $a$ could help me solve this question.
70652
Numerically it seems to be true that  $$ \int_0^\infty \frac{\sin x}{\sqrt{x}}dx=\sqrt{\frac{\pi}{2}}. $$  Any ideas how to prove this?
95551
I'd like some help with the following integral:  $$\int_0^\infty \lfloor x \rfloor e^{-x}\mathrm dx .$$  Thanks.