Category: Integration

Question Number 207582 by sniper237 last updated on 19/May/24 $${\int }_0^{\pi }ln\left(sinx\right)dx=-\pi ln2$$$${\int }_0^{1}ln\mathrm{\Gamma }\left(x\right)dx=ln\left(2\pi \right)$$ Answered by mathzup last updated on 21/May/24…

Question Number 207424 by MetaLahor1999 last updated on 14/May/24 $$f_n\left(x\right):=\int e^{\frac{2x}{3}}\frac{cos\left(x\right)}{{(cos\left(x\right)+sin\left(x\right))}^{\frac{n}{3}}}dx=\dots ?$$$$forn=1,ifound$$$$f_1\left(x\right)=\frac{3}{4}{e}^{\frac{2x}{3}}{(cos\left(x\right)+sin\left(x\right))}^{\frac{2}{3}}+C$$$$isthereanyideasforageneralcaseor$$$${the}\:{case}\:{n}=\mathrm{2}? \…

Question Number 207382 by efronzo1 last updated on 13/May/24 Answered by sniper237 last updated on 13/May/24 $$\overset{{X}=^{\mathrm{3}} \sqrt{{x}−\mathrm{2}}} {=}\underset{{X}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{X}^{\mathrm{6}} +\mathrm{2}{X}^{\mathrm{3}} +{X}}{\:^{\mathrm{3}} \sqrt{\mathrm{4}−\mathrm{2}\sqrt{\mathrm{3}{X}^{\mathrm{3}} +\mathrm{4}}−{X}^{\mathrm{3}} \sqrt{\mathrm{3}{X}^{\mathrm{3}}…

Question Number 207359 by Shrodinger last updated on 12/May/24 $$\int \frac{ln(x^2+sin\left(sin\left(e^x\right)\right))}{\sqrt{x+tan\left(ln\left(x\right)\right)}}dx$$ Answered by Berbere last updated on 12/May/24 $$itsemmsnoncloseforme$$ Commented…

Question Number 207099 by tri26112004 last updated on 06/May/24 Answered by Berbere last updated on 06/May/24 $$={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{e^{i\pi ax}}{{(x^2+{\beta }^2)}^{n+1}}dx;a\in {\mathbb{R}}_{+}$$$${if}\:{Imx}\geqslant\mathrm{0}\:\mid{e}^{{i}\pi{ax}}…