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Category: Integration

x-2-1-x-5-dx-

Question Number 87503 by M±th+et£s last updated on 04/Apr/20 $$\int\frac{{x}^{\mathrm{2}} }{\mathrm{1}+{x}^{\mathrm{5}} }{dx} \\ $$ Commented by MJS last updated on 04/Apr/20 $${x}^{\mathrm{5}} +\mathrm{1}=\left({x}+\mathrm{1}\right)\left({x}^{\mathrm{2}} −\frac{\mathrm{1}−\sqrt{\mathrm{5}}}{\mathrm{2}}{x}+\mathrm{1}\right)\left({x}^{\mathrm{2}} −\frac{\mathrm{1}+\sqrt{\mathrm{5}}}{\mathrm{2}}+\mathrm{1}\right)…

e-1-e-1-lnx-2-x-dx-

Question Number 87461 by hamdhan last updated on 04/Apr/20 $$\underset{{e}^{-\mathrm{1}} } {\overset{\mathrm{e}} {\int}}\:\frac{\sqrt{\mathrm{1}−\left(\mathrm{ln}{x}\right)^{\mathrm{2}} }}{{x}}\:{dx} \\ $$ Commented by ajfour last updated on 04/Apr/20 $${let}\:\:\mathrm{ln}\:{x}={t}\:\:\:\:\Rightarrow\:\:\left({dx}\right)/{x}={dt} \\…

Q-If-a-b-are-positive-numbers-and-a-1-6a-2-1-3-b-1-6b-2-1-3-then-find-the-value-of-a-b-Compile

Question Number 152939 by mnjuly1970 last updated on 03/Sep/21 $$ \\ $$$$\:\:\:\:\:\mathrm{Q}\::\:\:\mathrm{If}\:\:\:\:{a}\:\:,\:\:{b}\:\:\:\:\mathrm{are}\:\mathrm{positive}\:\mathrm{numbers}\:\:\mathrm{and} \\ $$$$\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\begin{cases}{\:\:{a}\:=\:\mathrm{1}\:+\:\sqrt[{\mathrm{3}}]{\:\mathrm{6}{a}\:−\mathrm{2}}\:\:}\\{\:\:\:{b}\:=\:\mathrm{1}\:+\:\sqrt[{\mathrm{3}}]{\:\mathrm{6}{b}\:−\mathrm{2}}}\end{cases} \\ $$$$\:\:\:\:\:\:\:\:\:\mathrm{then}\:\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:,\:\:\:\:{a}.{b}\:=? \\ $$$$\:\:\:\:…\:\mathrm{Compiled}\:\mathrm{by}\:\mathrm{m}.\mathrm{n}\::\:\left(\mathscr{E}\:{lementary}\:{olympiad}\:\right).\:\:\:\:\:\:\blacksquare \\ $$$$ \\ $$ Answered…

cos-x-5-4cos-x-2-dx-

Question Number 87340 by john santu last updated on 04/Apr/20 $$\int\:\frac{\mathrm{cos}\:\mathrm{x}}{\left(\mathrm{5}+\mathrm{4cos}\:\mathrm{x}\right)^{\mathrm{2}} }\:\mathrm{dx}\:= \\ $$ Commented by john santu last updated on 04/Apr/20 $$\mathrm{dear}\:\mathrm{mr}\:\mathrm{Mjs}.\:\mathrm{whether}\:\mathrm{methods}\: \\ $$$$\mathrm{besides}\:\mathrm{weierstrass}\:\mathrm{substitution}?…