Category: Integration

Question Number 67008 by mathmax by abdo last updated on 21/Aug/19 $$letf\left(x\right)={\int }_0^{\mathrm{\infty }}\frac{dt}{{(x^2+t^2)}^2}withx>0$$$$1)findaexplicitformof\left(x\right)$$$$\left.\mathrm{2}\right){find}\:{also}\:{g}\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dt}}{\left({x}^{\mathrm{2}} \:+{t}^{\mathrm{2}} \right)^{\mathrm{3}}…

Question Number 67006 by mathmax by abdo last updated on 21/Aug/19 $$calculae{A}_n={\int }_0^{\mathrm{\infty }}\frac{dx}{{(x^2+1)}^n}withnintegrnaturalandn>0$$ Commented by mathmax by abdo last…

Question Number 132534 by liberty last updated on 15/Feb/21 $${\int }_0^{\frac{\pi }{2}}\frac{\mathrm{dx}}{1+\sin \mathrm{x}}\to \mathrm{diverges}\mathrm{or}\mathrm{converges}?$$ Commented by MJS_new last updated on 15/Feb/21 $$\mathrm{we}\mathrm{only}\mathrm{want}\mathrm{to}\mathrm{know}\mathrm{if}\mathrm{it}\mathrm{converges}\mathrm{or}\mathrm{not}$$$$\mathrm{0}\leqslant{x}\leqslant\frac{\pi}{\mathrm{2}}\:\Rightarrow\:\frac{\mathrm{1}}{\mathrm{2}}\leqslant\frac{\mathrm{1}}{\mathrm{1}+\mathrm{sin}\:{x}}\leqslant\mathrm{1}\:\Rightarrow\:\mathrm{converges} \…

Question Number 1444 by 112358 last updated on 04/Aug/15 $$Evaluatethefollowingintegral.$$$${\int }_0^n\lfloor x{\rfloor }^{1/\lfloor x\rfloor !}dx(n\in \mathbb{N})$$$$Here\lfloor x\rfloor istheinteger-partofx$$$$e.g\lfloor 0.12\rfloor =0,\lfloor 5.896\rfloor =5$$$$$$ Answered by…

Question Number 132473 by physicstutes last updated on 14/Feb/21 $$\int \frac{x\cosh x}{{\left(\sinh x\right)}^2}dx$$ Answered by mathmax by abdo last updated on 14/Feb/21 $$\mathrm{I}=\int\:\frac{\mathrm{xchx}}{\mathrm{sh}^{\mathrm{2}} \mathrm{x}}\mathrm{dx}\:\:\mathrm{by}\:\mathrm{parts}\:\:\mathrm{u}^{'} \:=\frac{\mathrm{chx}}{\mathrm{sh}^{\mathrm{2}}…