# Evaluation of an improper integral: $\int_0^\infty \frac{t\sin{t}}{u^2+t^2\,}dt$

I want to calculate the integral below for positive $u$. $$\int_0^\infty \dfrac{t\sin{t}}{u^2+t^2}dt$$ Wolframalpha gives me a simple answer : $\dfrac{\pi}{2}e^{-u}$.

but I cannot approach to that. Can anyone solve above without using complex integral (ex.residue integration.. because I'm beginner of analysis)?

Thanks.

• ${1 \over 2}\,\pi\exp\left(-\left\vert u\right\vert\right)$. – Felix Marin Nov 8 '16 at 3:49

## 1 Answer

Hint. Let's consider the Laplace transform of $\displaystyle I(a):=\int_{0}^{\infty}\frac{\cos(ax)}{x^2+1}\:dx$. We have \begin{aligned} \mathcal{L}\left(I(a)\right)(s)&=\mathcal{L}\left(\int_{0}^{\infty}\frac{\cos(ax)}{x^2+1}\:dx\right)(s) \\& = \int_{0}^{\infty}\int_{0}^{\infty}\frac{\cos(ax)}{x^2+1}e^{-as}\:da\:dx \\&= \int_{0}^{\infty}\frac{s}{(x^2+1)(s^2+x^2)}\;{dx} \\&= \frac{\pi}{2(s+1)} \end{aligned}giving $$I(a)=\int_{0}^{\infty}\frac{\cos(ax)}{x^2+1}\:dx=\mathcal{L}^{-1}\left(\frac{\pi}{2(s+1)}\right) =\frac{\pi}{2}e^{-a},\qquad a>0, \tag2$$ then by differentiating $(2)$ with respect to $a$, one gets $$\int_{0}^{\infty}\frac{x\sin(ax)}{x^2+1}\:dx=\frac{\pi}{2}e^{-a},\qquad a>0. \tag3$$ as given by Wolfram alpha.

• Understand it, but how can I justify the exchange of improper integral between second and third equalities?? – Jinmu You Nov 7 '16 at 17:13
• @JinmooYou This may be justified by the following inequalities: $\left|\int_{0}^{\infty}\frac{\cos(ax)}{x^2+1}\:dx \right|\le \int_{0}^{\infty}\frac1{x^2+1}\:dx<\infty$ and $\left|\int_{0}^{\infty}\cos(ax)e^{-as}\:da\right|\le \int_{0}^{\infty}1\cdot e^{-as}\:da<\infty$ which give the right to use Fubini's theorem. – Olivier Oloa Nov 7 '16 at 19:52
{"ft_lang_label":"__label__en","ft_lang_prob":0.77183884,"math_prob":0.99999607,"size":320,"snap":"2019-35-2019-39","text_gpt3_token_len":92,"char_repetition_ratio":0.07594936,"word_repetition_ratio":0.0,"special_character_ratio":0.28125,"punctuation_ratio":0.140625,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.0000035,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-08-24T03:01:09Z\",\"WARC-Record-ID\":\"<urn:uuid:ac1dfb5a-87bb-45cd-8ae2-f07cb43e5836>\",\"Content-Length\":\"138388\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:8820c1ba-9bff-4401-ac06-1615bde4cd94>\",\"WARC-Concurrent-To\":\"<urn:uuid:66fa8bae-be42-4564-a277-757b211af09e>\",\"WARC-IP-Address\":\"151.101.65.69\",\"WARC-Target-URI\":\"https://math.stackexchange.com/questions/2003796/evaluation-of-an-improper-integral-int-0-infty-fract-sintu2t2-dt\",\"WARC-Payload-Digest\":\"sha1:EIXW6VPGY7HHHJDFMRWHVGA3MA5II3QC\",\"WARC-Block-Digest\":\"sha1:NEAAMCE75OFJM5EEGERJDBY6I64KH3JD\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-35/CC-MAIN-2019-35_segments_1566027319470.94_warc_CC-MAIN-20190824020840-20190824042840-00527.warc.gz\"}"}