Category: Integration

Question Number 50413 by Abdo msup. last updated on 16/Dec/18 $$letf\in C^0(R,R)/\mathrm{\forall }x\in Rf(a+b-x)=f\left(x\right)$$$$1)find{\int }_a^{b}xf\left(x\right)dxintermsof{\int }_a^{b}f\left(x\right)dx$$$$2)calculate{\int }_0^{\pi }\frac{xdx}{1+sinx}$$ Commented…

Question Number 50412 by Abdo msup. last updated on 16/Dec/18 $$1)calculate{U}_n={\int }_0^{\pi }\frac{dx}{1+{cos}^2\left(nx\right)}withnfromN$$$$2)fcontinuefrom[0,\pi ]toRfind$$$${lim}_{n\to +\mathrm{\infty }}{\int }_0^{\pi }\frac{f\left(x\right)}{1+{cos}^2\left(nx\right)}dx$$…

Question Number 50407 by Abdo msup. last updated on 16/Dec/18 $$determinef\in C^0([0,1],R)verifying$$$${\int }_0^{1}f\left(x\right)dx=\frac{1}{3}+{\int }_0^1{\left(f\left(x^2\right)\right)}^{2}dx$$ Terms of Service…

Question Number 115927 by mathmax by abdo last updated on 29/Sep/20 $$\mathrm{calculate}{\int }_1^{+\mathrm{\infty }}\frac{\mathrm{dx}}{{({4\mathrm{x}}^2-1)}^3}$$ Answered by 1549442205PVT last updated on 29/Sep/20…

Question Number 115920 by mnjuly1970 last updated on 29/Sep/20 $$$$$$provethat::$$$$$$$${\int }_0^{\mathrm{\infty }}({tanh}^a\left(x\right)-{tanh}^b\left(x\right))dx$$$$\stackrel{???}{=}\frac{\psi \left(\frac{b+1}{2}\right)-\psi \left(\frac{a+1}{2}\right)}{2}$$$$\:\:\:\:\:\:\:\:\:\:{m}.{n}.{july}.\mathrm{1970}…