Question Number 88170 by Ar Brandon last updated on 08/Apr/20
$${Prove}\:{that}\: \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} {tcos}\:{n}\pi{tdt}=\frac{\left(−\mathrm{1}\right)^{{n}} −\mathrm{1}}{{n}^{\mathrm{2}} \pi^{\mathrm{2}} } \\ $$
Commented by jagoll last updated on 08/Apr/20
![= ((t.sin nπt)/(nπ)) + ((cos nπt)/(n^2 π^2 )) ] _0^1 = ((sin nπ)/(nπ)) + ((cos nπ)/(n^2 π^2 )) − (1/(n^2 π^2 )) = (((−1)^n −1)/(n^2 π^2 ))](https://www.tinkutara.com/question/Q88172.png)
$$\left.=\:\frac{\mathrm{t}.\mathrm{sin}\:\mathrm{n}\pi\mathrm{t}}{\mathrm{n}\pi}\:+\:\frac{\mathrm{cos}\:\mathrm{n}\pi\mathrm{t}}{\mathrm{n}^{\mathrm{2}} \pi^{\mathrm{2}} }\:\right]\:_{\mathrm{0}} ^{\mathrm{1}} \\ $$$$=\:\frac{\mathrm{sin}\:\mathrm{n}\pi}{\mathrm{n}\pi}\:+\:\frac{\mathrm{cos}\:\mathrm{n}\pi}{\mathrm{n}^{\mathrm{2}} \pi^{\mathrm{2}} }\:−\:\frac{\mathrm{1}}{\mathrm{n}^{\mathrm{2}} \pi^{\mathrm{2}} } \\ $$$$\:=\:\:\frac{\left(−\mathrm{1}\right)^{\mathrm{n}} −\mathrm{1}}{\mathrm{n}^{\mathrm{2}} \pi^{\mathrm{2}} } \\ $$
Commented by Joel578 last updated on 08/Apr/20
$$\mathrm{for}\:{n}\:\in\:\mathbb{N} \\ $$