Category: Integration

Question Number 40151 by maxmathsup by imad last updated on 16/Jul/18 $$letF\left(x\right)={\int }_0^{\frac{\pi }{2}}cos\left(xsint\right)dt$$$$1)provethat\mathrm{\forall }u\in R1-\frac{u^2}{2}⩽cosu⩽1-\frac{u^2}{2}+\frac{u^4}{24}$$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\frac{\pi}{\mathrm{2}}\left(\mathrm{1}−\frac{{x}^{\mathrm{2}} }{\mathrm{4}}\right)\leqslant{F}\left({x}\right)\leqslant\:\frac{\pi}{\mathrm{2}}\left(\mathrm{1}−\frac{{x}^{\mathrm{2}} }{\mathrm{4}}\:+\frac{{x}^{\mathrm{4}} }{\mathrm{64}}\right) \…

Question Number 40148 by maxmathsup by imad last updated on 16/Jul/18 $$letf\left(x\right)=\frac{x^3}{{(1+x^2)}^{\frac{3}{2}}}$$$$1)calculate{\int }_0^{1}f\left(x\right)dx$$$$\left.\mathrm{2}\right)\:{let}\:\:{S}_{{n}} =\:\frac{\mathrm{1}}{{n}^{\mathrm{4}} }\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\:\:\:\frac{{k}^{\mathrm{3}}…

Question Number 40147 by maxmathsup by imad last updated on 16/Jul/18 $$calculate{\int }_0^2\sqrt{x^3(2-x)}dx$$ Commented by math khazana by abdo last updated…

Question Number 40146 by maxmathsup by imad last updated on 16/Jul/18 $$find{\int }_{\frac{1}{2}}^1\frac{dx}{\sqrt{4x^2-1}+\sqrt{4x^2+1}}$$ Commented by maxmathsup by imad last updated…

Question Number 40144 by maxmathsup by imad last updated on 16/Jul/18 $$find{\int }_1^{2}x\sqrt{x^2-2x+5}dx$$ Answered by maxmathsup by imad last updated on…