Category: Integration

Question Number 118924 by mnjuly1970 last updated on 21/Oct/20 $$\dots advancedcalculus\dots$$$$provethat::$$$$\underset{n=1}{\sum ^{\mathrm{\infty }}}\frac{{(-1)}^n{H}_n}{n^2}={\int }_0^1\frac{ln(1-x)ln(1+x)}{x}dx$$$$\:\:\:\:\:{note}\:::\:{H}_{{n}} =\underset{{k}=\mathrm{1}} {\overset{{n}}…

Question Number 118905 by benjo_mathlover last updated on 20/Oct/20 $$\int \frac{d\lambda }{{({\lambda }^2-9)}^2}=?$$ Answered by Bird last updated on 21/Oct/20 $${I}=\int\:\frac{{dx}}{\left({x}^{\mathrm{2}} −\mathrm{9}\right)^{\mathrm{2}} }\:\Rightarrow{I}=\int\:\frac{{dx}}{\left({x}−\mathrm{3}\right)^{\mathrm{2}} \left({x}+\mathrm{3}\right)^{\mathrm{2}}…

Question Number 118886 by mnjuly1970 last updated on 20/Oct/20 $$\dots advancedcalculus\dots$$$$$$$$\:\:\:\:\:\:\:\:{evaluate}\:::\:\:\left\{_{\mathrm{2}.\:\Omega_{\mathrm{2}} =\:\int_{\mathrm{0}} ^{\:\frac{\mathrm{1}}{\mathrm{2}}} \:\frac{{ln}^{\mathrm{2}} \left(\mathrm{1}+{x}\right)}{{x}}\:{dx}=??} ^{\mathrm{1}.\:\Omega_{\mathrm{1}} =\int_{\mathrm{0}} ^{\:\frac{\mathrm{1}}{\mathrm{2}}} \:\frac{{ln}^{\mathrm{2}} \left(\mathrm{1}−{x}\right)}{{x}}{dx}=??} \right. \…

Question Number 53295 by gunawan last updated on 20/Jan/19 $${\int }_0^{\frac{\pi }{2}}\frac{1}{2+\cos x}dx=\dots$$ Commented by maxmathsup by imad last updated on 20/Jan/19 $${changement}\:{tan}\left(\frac{{x}}{\mathrm{2}}\right)={t}\:{give}\: \…

Question Number 53294 by gunawan last updated on 20/Jan/19 $${\int }_{-1/2}^{1/2}{[{\left(\frac{x+1}{x-1}\right)}^2+{\left(\frac{x-1}{x+1}\right)}^2-2]}^{1/2}dx=\dots$$ Answered by tanmay.chaudhury50@gmail.com last updated on 20/Jan/19 $${f}\left({x}\right)=\left[\left(\frac{{x}+\mathrm{1}}{{x}−\mathrm{1}}\right)^{\mathrm{2}}…