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Category: Integration

x-4-1-x-4-dx-

Question Number 84106 by jagoll last updated on 09/Mar/20 $$\int\:\frac{\mathrm{x}^{\mathrm{4}} }{\:\sqrt{\mathrm{1}−\mathrm{x}^{\mathrm{4}} }}\:\mathrm{dx}\:=\:? \\ $$ Commented by MJS last updated on 09/Mar/20 $${t}=\sqrt{\mathrm{sin}\:{x}}\:\rightarrow\:{dx}=\frac{\sqrt{\mathrm{1}−{x}^{\mathrm{4}} }}{\mathrm{2}{x}}{dt} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}\int\left(\mathrm{sin}\:{t}\right)^{\frac{\mathrm{3}}{\mathrm{2}}}…

Find-the-reduction-formula-x-n-e-ax-dx-

Question Number 84101 by niroj last updated on 09/Mar/20 $$\:\boldsymbol{\mathrm{Find}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{reduction}}\:\boldsymbol{\mathrm{formula}} \\ $$$$\:\:\int\boldsymbol{\mathrm{x}}^{\boldsymbol{\mathrm{n}}} \boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{ax}}} \:\boldsymbol{\mathrm{dx}} \\ $$ Answered by MJS last updated on 09/Mar/20 $$\int{x}^{{n}} \mathrm{e}^{{ax}}…

1-0-2pi-1-a-sin-t-dt-a-gt-0-2-2pi-4pi-1-2-sin-t-dt-

Question Number 149625 by puissant last updated on 06/Aug/21 $$\left.\mathrm{1}\right)\int_{\mathrm{0}} ^{\mathrm{2}\pi} \frac{\mathrm{1}}{\mathrm{a}+\mathrm{sin}\left(\mathrm{t}\right)}\mathrm{dt}\:,\:\mathrm{a}>\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\int_{\mathrm{2}\pi} ^{\mathrm{4}\pi} \frac{\mathrm{1}}{\mathrm{2}+\mathrm{sin}\left(\mathrm{t}\right)}\mathrm{dt}.. \\ $$ Answered by ArielVyny last updated on 06/Aug/21…

K-1-1-sin-2-x-dx-

Question Number 149608 by puissant last updated on 06/Aug/21 $$…..\mathrm{K}=\int\frac{\mathrm{1}}{\mathrm{1}+\mathrm{sin}^{\mathrm{2}} \left(\mathrm{x}\right)}\mathrm{dx}…… \\ $$ Answered by ArielVyny last updated on 06/Aug/21 $${sin}^{\mathrm{2}} {x}=\frac{\mathrm{1}−{cos}\left(\mathrm{2}{x}\right)}{\mathrm{2}} \\ $$$${K}=\int\frac{\mathrm{1}}{\mathrm{1}+\frac{\mathrm{1}−{cos}\left(\mathrm{2}{x}\right)}{\mathrm{2}}}=\int\frac{\mathrm{1}}{\frac{\mathrm{2}+\mathrm{1}−{cos}\left(\mathrm{2}{x}\right)}{\mathrm{2}}}{dx} \\…

Please-any-short-cut-to-evaluate-1-2-x-x-4-1-3-x-4-

Question Number 84046 by TawaTawa1 last updated on 09/Mar/20 $$\mathrm{Please}\:\mathrm{any}\:\mathrm{short}\:\mathrm{cut}\:\mathrm{to}\:\mathrm{evaluate}:\:\:\:\:\:\int_{\:\mathrm{1}} ^{\:\mathrm{2}} \:\:\:\frac{\:\:\sqrt[{\mathrm{3}}]{\mathrm{x}\:\:−\:\:\mathrm{x}^{\mathrm{4}} }}{\mathrm{x}^{\mathrm{4}} } \\ $$ Commented by jagoll last updated on 09/Mar/20 $$\mathrm{i}\:\mathrm{don}'\mathrm{t}\:\mathrm{know}\:\mathrm{short}\:\mathrm{cut}\:\mathrm{method} \\…

x-4-x-4-1-dx-

Question Number 84035 by M±th+et£s last updated on 08/Mar/20 $$\int\frac{{x}−\mathrm{4}}{{x}^{\mathrm{4}} −\mathrm{1}}{dx} \\ $$ Commented by MJS last updated on 08/Mar/20 $$\frac{{x}−\mathrm{4}}{{x}^{\mathrm{4}} −\mathrm{1}}=−\frac{{x}}{\mathrm{2}\left({x}^{\mathrm{2}} +\mathrm{1}\right)}+\frac{\mathrm{2}}{{x}^{\mathrm{2}} +\mathrm{1}}+\frac{\mathrm{5}}{\mathrm{4}\left({x}+\mathrm{1}\right)}−\frac{\mathrm{3}}{\mathrm{4}\left({x}−\mathrm{1}\right)} \\…