I want to see the method for computing the following integrals: $$\int_0^\infty\ln(x)\sin(x^2)dx$$ and $$\int_0^\infty\ln(x)\cos(x^2)dx$$ I believe I have seen these in this forum before, but I cannot find them again using the search function. Can someone help me compute these integrals, or maybe just share the original link? Thank you.
120467
$$problem:\int_0^\infty cos(x^2)\ln(x)dx$$ $$Re\int_0^\infty e^{-ix^2}ln(x)dx$$ $$\Gamma(s)=\int_0^\infty x^{s-1}e^dx, \ \ \ \ x=u^2, dx=2udu$$ $$2\int_0^\infty u^{2s-1}e^{-u^2}du$$ $$\Gamma'(s)=4\int_0^\infty u^{2s-1}e^{-u^2}ln(u)du$$ $$\frac{\Gamma'(1/2)}{4}=\int_0^\infty e^{-u^2}ln(u)du$$ $$Re\int_0^\infty e^{-ix^2}ln(x)dx$$ $$Re\frac{1}{\sqrt{i}}\int e^{-u^2}ln(u)-\frac{1}{\sqrt{i}}\int e^{-u^3}ln(\sqrt{i})du$$ $$Re[\frac{\Gamma'(\frac12)}{4\sqrt{i}}-\frac{\sqrt{\pi}ln(i)}{4\sqrt{i}}]=Re[\frac{\Gamma'\frac(12)-\sqrt{pi}\ln(i)}{4\sqrt{i}}]$$ $$Re[\frac{\Gamma(\frac12)\psi(\frac12)-\sqrt{\pi}\ln(i)}{4\sqrt{i}}]$$ $$Re[\frac{\sqrt{pi}(-\gamma-2\ln(2))-\frac{\pi^{\frac32}}{2}i}{4\sqrt{i}}]$$ $$=\frac{\sqrt{2\pi}}{8}(-\gamma-2\ln(2))-\frac{\pi^{\frac32}}{16}$$
74195
Good afternoon, I need to find:  $$\int_0^\infty\frac{x^n}{ax^2+bx+c}\space dx$$  I've no idea about any clue what so ever.
59265
I've been wondering, what is the general form for the integral of $\sin(\cos x)$? I tried integrating it myself and couldn't, tried with Wolfram Alpha and it couldn't either. If you think about it conceptually, it should have a general form as it's graph if just like sin and cos, which both have general forms. If it doesn't have a general form, I'd like to know why. Thanks!
89151
I am practicing calculus section of GRE Math Subject test, and I can't figure out a way to do the following integral: $$\int_0^\infty \frac{\cos (ax) - \cos (bx)}{x}dx.$$  I absolutely have no clue how to do this. Can someone show me the solution explicitly? Thank you.
91157
Question:   Compute $$\int _0 ^{2\pi} \left(\sum_{n=1}^\infty \frac {\cos(nx)}{2^n}\right)^2 dx$$   Thoughts: Tried interchanging the integral and the sum, but then the integral turned out to be zero...
55145
I'm taking a complex analysis course and doing some practice computing residues &amp; evaluating integrals. I pulled out an old book called     On page 196-197, there are some interesting integrals to evaluate. I'm at 5.4.3.10.: I was able to do question 1, but was stumped at how to even begin with question 2:     Evaluate the integral   $$\int_0^{2 \pi} e^{\cos x} \cos (nx - \sin x) \,dx ,$$   where $n$ is an natural number.   The answer is simply $\frac{2 \pi}{n}$. Any hints?
75155
So I came across this question on brilliant.org and didn't know how to go about it: $$\int_{0}^{\infty} \frac{ \ln(x)}{x^2+2x+4} dx $$  I tried to complete the squares in the denominator and then use a trigonometric substitution (with tan) but I didn't get anything beyond that.   How do I do it? Do we need to differentiate under the integral sign here?
92294
Calculate the integral $$\int \ln (\sin x) \, dx.$$
114157
How does one evaluate a function in the form of $$\int \ln^nx\space dx$$ My trusty friend Wolfram Alpha is blabbering about $\Gamma$ functions and I am having trouble following. Is there a method for indefinitely integrating such and expression? Or if there isn't a method how would you tackle the problem?
23165
I would like your help with deciding whether the following integral converges or not:  $$\int_{0}^{\infty}\frac{dx}{1+(x\sin5x)^2}.$$  I tried to compare it to other functions and to change the variables, but it didn't work for me.  Thanks a lot!
74829
I have seen $$\int_0^\infty \frac{\cos(x)}{x^2+1} \, dx=\frac{\pi}{2e}$$ evaluated in various ways.   It's rather popular when studying CA.   But, what about $$\int_0^\infty \frac{\sin(x)}{x^2+1} \, dx\,\,?$$  This appears to be trickier and more challenging.  I found that it has a closed form of  $$\cosh(1)\operatorname{Shi}(1)-\sinh(1)\text{Chi(1)}\,\,,\,\operatorname{Shi}(1)=\int_0^1 \frac{\sinh(x)}{x}dx\,\,,\,\, \text{Chi(1)}=\gamma+\int_0^1 \frac{\cosh(x)-1}{x} \, dx$$  which are the hyperbolic sine and cosine integrals, respectively.  It's an odd function, so  $$\int_{-\infty}^\infty \frac{\sin(x)}{x^2+1} \, dx=0$$  But, does anyone know how the former case can be done? Thanks a bunch.
54941
$$ \int_0^{\infty } \frac{\log (x)}{e^x+1} \, dx = -\frac{1}{2} \log ^2(2) $$  Anyone an idea on how to prove this?
35050
I have to calculate $$\int_0^\infty\frac{\log x dx}{x^2-1},$$  and the hint is to integrate $\frac{\log z}{z^2-1}$ over the boundary of the domain $$\{z\,:\,r&lt;|z|&lt;R,\,\Re (z)&gt;0,\,\Im (z)&gt;0\}.$$  I don't understand. The boundary of this domain has a pole of the integrand in it, doesn't it? Doesn't it make this method useless?
113571
Compute  $$ \int_2^4 \frac{ \sqrt{\ln(9-x)}      }{ \sqrt{\ln(9-x)}+\sqrt{\ln(x+3)}        } dx. $$ I am not sure how to start this one...I am thinking of a substitution to get started.
72077
How to calculate the following integral? $$\int_{-\infty}^\infty\dfrac{\omega^\alpha e^{i\omega t}}{(\omega_0^2-\omega^2)^{2}+4(\zeta\omega_0\omega)^2}\,d\omega$$  where $\alpha&gt;0,t&gt;0,\zeta&gt;0,\omega_0&gt;0$.  When $\alpha=0$, I can figure it out by Residue theorem. However, I do not have any idea how to calculate this integral when $\alpha&gt;0$. Does this integral have an analytic solution?
19389
I have to compute this integral $$\int_{0}^{\infty} \frac{dt}{1+t^4}$$ to solve a problem in a homework. I have tried in many ways, but I'm stuck. A search in the web reveals me that it can be do it by methods of complex analysis. But I have not taken this course yet. Thanks for any help.
39954
Calculate  $$\int^{\infty}_{0}\frac{\ln^2(x)}{(1-x^2)^2}dx$$   I have tried to put $\displaystyle x=\frac{1}{t}$ and $\displaystyle dx=-\frac{1}{t^2}dt$  $$ \int^{\infty}_{0}\frac{t^2\ln^2(t)}{(t^2-1)^2}dt$$  $$\frac{1}{2}\int^{\infty}_{0}t\ln^2(t)\frac{2t}{(t^2-1)^2}dt$$  $$ \frac{1}{2}\bigg[-t\ln^2(t)\frac{1}{t^2-1}+\int^{\infty}_{0}\frac{\ln^2(t)}{t^2-1}+2\int^{\infty}_{0}\frac{\ln(t)}{t^2-1}dt\bigg]$$  How can I solve it?