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Category: Integration

I-0-lnx-x-2-2x-4-dx-put-x-1-3-tan-I-0-pi-2-ln-3-tan-1-3-sec-2-3-sec-2-d-I-1-3-0-pi-2-ln-3-tan-1-d-I-1-3-0-pi-2-ln-3-sin-cos-

Question Number 118934 by Riteshgoyal last updated on 20/Oct/20 $$ \\ $$$$ \\ $$$${I}=\int_{\mathrm{0}} ^{\infty} \:\frac{{lnx}}{{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{4}}\:{dx} \\ $$$${put}\:{x}+\mathrm{1}=\sqrt{\mathrm{3}}\:{tan}\theta \\ $$$${I}=\int_{\mathrm{0}} ^{\pi/\mathrm{2}} \:\frac{{ln}\left(\sqrt{\mathrm{3}}\:{tan}\theta−\mathrm{1}\right)}{\mathrm{3}\left({sec}^{\mathrm{2}} \theta\right)}\:\sqrt{\mathrm{3}}\:{sec}^{\mathrm{2}} \theta{d}\theta…

advanced-calculus-prove-that-n-1-1-n-H-n-n-2-0-1-ln-1-x-ln-1-x-x-dx-note-H-n-k-1-n-1-k-therefore-n-1-

Question Number 118924 by mnjuly1970 last updated on 21/Oct/20 $$\:\:\:\:\:\:\:\:\:…\:{advanced}\:{calculus}… \\ $$$$\:\:\:\:{prove}\:{that}\::: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} {H}_{{n}} }{{n}^{\mathrm{2}} }\:=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{ln}\left(\mathrm{1}−{x}\right){ln}\left(\mathrm{1}+{x}\right)\:\:}{{x}}{dx}\:\: \\ $$$$\:\:\:\:\:{note}\:::\:{H}_{{n}} =\underset{{k}=\mathrm{1}} {\overset{{n}}…

if-u-e-xyz-then-u-xyx-a-u-xyz-2-3xyz-1-b-u-3-xyz-2-1-c-u-xyz-2-2yz-1-please-help-

Question Number 53378 by Necxx last updated on 21/Jan/19 $${if}\:{u}={e}^{{xyz}} \:{then}\:{u}_{{xyx}} =? \\ $$$$\left.{a}\left.\right){u}\left(\left({xyz}\right)^{\mathrm{2}} +\mathrm{3}{xyz}+\mathrm{1}\right)\:{b}\right){u}\left(\mathrm{3}\left({xyz}\right)^{\mathrm{2}} +\mathrm{1}\right) \\ $$$$\left.{c}\right){u}\left(\left({xyz}\right)^{\mathrm{2}} +\mathrm{2}{yz}+\mathrm{1}\right) \\ $$$$ \\ $$$${please}\:{help} \\ $$…

d-2-9-2-

Question Number 118905 by benjo_mathlover last updated on 20/Oct/20 $$\:\:\:\int\:\frac{{d}\lambda}{\left(\lambda^{\mathrm{2}} −\mathrm{9}\right)^{\mathrm{2}} }\:=?\: \\ $$ 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}}…

advanced-calculus-evaluate-2-2-0-1-2-ln-2-1-x-x-dx-1-1-0-1-2-ln-2-1-x-x-dx-M-N-1970-

Question Number 118886 by mnjuly1970 last updated on 20/Oct/20 $$\:\:\:\:\:\:\:\:\:\:…\:\:{advanced}\:\:{calculus}… \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:{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. \\…

x-4-1-x-5-4x-3-dx-

Question Number 118834 by bramlexs22 last updated on 20/Oct/20 $$\:\:\int\:\frac{{x}^{\mathrm{4}} +\mathrm{1}}{{x}^{\mathrm{5}} +\mathrm{4}{x}^{\mathrm{3}} }\:{dx}\: \\ $$ Answered by bobhans last updated on 20/Oct/20 $$\:{Solve}\:\int\:\frac{{x}^{\mathrm{4}} +\mathrm{1}}{{x}^{\mathrm{5}} +\mathrm{4}{x}^{\mathrm{3}}…