Category: Integration

Question Number 129764 by Eric002 last updated on 18/Jan/21 $$provethat$$$${\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{x}^2{e}^{-x^2}cos\left(x^2\right)sin\left(x^2\right)dx$$$$=\frac{\sqrt{\pi }sin\left[\frac{\sqrt{3}{tan}^{-1}\left(2\right)}{2}\right]}{4\sqrt[4]{125}}$$ Answered…

Question Number 64224 by mmkkmm000m last updated on 16/Jul/19 $$\int x_{x}dx$$$$$$$$\int x^{x}dx$$ Commented by mathmax by abdo last updated…

Question Number 129746 by bramlexs22 last updated on 18/Jan/21 $$\mathrm{N}=\int \frac{3+2\cos \mathrm{x}}{2+3\cos \mathrm{x}}\mathrm{dx}$$ Answered by liberty last updated on 18/Jan/21 $$\mathrm{N}=\int \frac{\frac{2}{3}(2+3\cos \mathrm{x})+\frac{8}{3}}{2+3\cos \mathrm{x}}\mathrm{dx}$$$$\mathrm{N}=\frac{2}{3}\mathrm{x}+\frac{8}{3}\int \frac{\mathrm{dx}}{2+3(2\cos {}^2\frac{\mathrm{x}}{2}-1)}$$$$\:\mathrm{N}=\:\frac{\mathrm{2}}{\mathrm{3}}\mathrm{x}+\frac{\mathrm{8}}{\mathrm{3}}\int\:\frac{\mathrm{dx}}{\mathrm{6cos}\:^{\mathrm{2}}…

Question Number 64176 by aliesam last updated on 15/Jul/19 Answered by MJS last updated on 15/Jul/19 $$\mathrm{the}\mathrm{minimum}\mathrm{of}f\left(x\right)=\sqrt{1+x^2}\mathrm{is}f\left(0\right)=1\Rightarrow \mathrm{the}$$$$\mathrm{area}\mathrm{between}\mathrm{the}x-\mathrm{axis}\mathrm{and}f\left(x\right)\mathrm{in}[-1;1]$$$$\mathrm{is}\:\mathrm{greater}\:\mathrm{than}\:\mathrm{2}\:\Rightarrow\:\mathrm{2}<\underset{−\mathrm{1}} {\int}^{\mathrm{1}} \sqrt{\mathrm{1}+{x}^{\mathrm{2}} }{dx}…

Question Number 64160 by mathmax by abdo last updated on 14/Jul/19 $$letf\left(x\right)={\int }_0^1\frac{dt}{x+2^t}withx>0$$$$1)determineaexplicitformforf\left(x\right)$$$$2)determinealsog\left(x\right)={\int }_0^1\frac{dt}{{(x+2^x)}^2}$$$$\left.\mathrm{3}\right)\:{give}\:{f}^{\left({n}\right)}…