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Category: Integration

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Question Number 47851 by maxmathsup by imad last updated on 15/Nov/18 $${let}\:\:{f}\left({x}\right)={x}+\mathrm{1}+\sqrt{{x}}\:{and}\:{g}\left({x}\right)={x}+\mathrm{1}−\sqrt{{x}} \\ $$$${find}\:\int\:\frac{{f}\left({x}\right)}{{g}\left({x}\right)}{dx}\:\:{and}\:\:\frac{\int{f}\left({x}\right){dx}}{\int{g}\left({x}\right){dx}}\:. \\ $$ Commented by maxmathsup by imad last updated on 16/Nov/18…

Question-178922

Question Number 178922 by mnjuly1970 last updated on 22/Oct/22 Answered by ARUNG_Brandon_MBU last updated on 22/Oct/22 $$\frac{\mathrm{1}}{{n}\left({n}+\mathrm{1}\right)\left({n}+\mathrm{2}\right)}=\frac{\frac{\mathrm{1}}{\mathrm{2}}}{{n}}−\frac{\mathrm{1}}{{n}+\mathrm{1}}+\frac{\frac{\mathrm{1}}{\mathrm{2}}}{{n}+\mathrm{2}} \\ $$$$\left(\frac{\mathrm{1}}{{n}\left({n}+\mathrm{1}\right)\left({n}+\mathrm{2}\right)}\right)^{\mathrm{2}} =\frac{\mathrm{1}}{\mathrm{4}}\left(\frac{\mathrm{1}}{{n}}−\frac{\mathrm{2}}{{n}+\mathrm{1}}+\frac{\mathrm{1}}{{n}+\mathrm{2}}\right)^{\mathrm{2}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{4}}\left(\frac{\mathrm{1}}{{n}^{\mathrm{2}} }+\frac{\mathrm{4}}{\left({n}+\mathrm{1}\right)^{\mathrm{2}} }+\frac{\mathrm{1}}{\left({n}+\mathrm{2}\right)^{\mathrm{2}} }+\mathrm{2}\left(\frac{−\mathrm{2}}{{n}\left({n}+\mathrm{1}\right)}+\frac{−\mathrm{2}}{\left({n}+\mathrm{1}\right)\left({n}+\mathrm{2}\right)}+\frac{\mathrm{1}}{{n}\left({n}+\mathrm{2}\right)}\right)\right)…

Question-113323

Question Number 113323 by AbhishekBasnet last updated on 12/Sep/20 Answered by bemath last updated on 12/Sep/20 $$\mathrm{let}\:\mathrm{a}+\mathrm{bx}\:=\:\mathrm{u}\:,\:\mathrm{x}\:=\:\frac{\mathrm{u}−\mathrm{a}}{\mathrm{b}} \\ $$$$\int\:\left(\frac{\mathrm{u}−\mathrm{a}}{\mathrm{b}}\right)\mathrm{u}^{\mathrm{3}} \left(\frac{\mathrm{du}}{\mathrm{b}}\right)\:=\:\frac{\mathrm{1}}{\mathrm{b}^{\mathrm{2}} }\int\:\left(\mathrm{u}^{\mathrm{4}} −\mathrm{au}^{\mathrm{3}} \right)\mathrm{du} \\ $$$$=\frac{\mathrm{1}}{\mathrm{b}^{\mathrm{2}}…

Question-178832

Question Number 178832 by Spillover last updated on 22/Oct/22 Answered by ARUNG_Brandon_MBU last updated on 22/Oct/22 $${a}.\:\int\frac{\mathrm{y}\left(\mathrm{y}−\mathrm{8}\right)}{\left(\mathrm{y}−\mathrm{4}\right)^{\mathrm{2}} }{d}\mathrm{y}=\int\frac{\left(\left(\mathrm{y}−\mathrm{4}\right)+\mathrm{4}\right)\left(\left(\mathrm{y}−\mathrm{4}\right)−\mathrm{4}\right)}{\left(\mathrm{y}−\mathrm{4}\right)^{\mathrm{2}} }{d}\mathrm{y} \\ $$$$\:\:\:\:=\int\frac{\left(\mathrm{y}−\mathrm{4}\right)^{\mathrm{2}} −\mathrm{16}}{\left(\mathrm{y}−\mathrm{4}\right)^{\mathrm{2}} }{d}\mathrm{y}=\int\left(\mathrm{1}−\frac{\mathrm{16}}{\left(\mathrm{y}−\mathrm{4}\right)^{\mathrm{2}} }\right){d}\mathrm{y} \\…