Category: Integration

Question Number 33689 by NECx last updated on 22/Apr/18 $$\int \frac{x}{x^3+1}dx$$ Commented by mondodotto@gmail.com last updated on 22/Apr/18 $$\mathbf{let}\mathbf{u}={\mathbf{x}}^3+1$$$$\frac{\boldsymbol{\mathrm{du}}}{\boldsymbol{\mathrm{dx}}}=\mathrm{3}\boldsymbol{\mathrm{x}}^{\mathrm{2}} \Rightarrow\boldsymbol{\mathrm{dx}}=\frac{\boldsymbol{\mathrm{du}}}{\mathrm{3}\boldsymbol{\mathrm{x}}^{\mathrm{2}}…

Question Number 99205 by bemath last updated on 19/Jun/20 Commented by bramlex last updated on 19/Jun/20 $$let\sqrt{4-\sqrt{x}}=t,4-\sqrt{x}=t^2$$$${(4-t^2)}^2=x;dx=-4t(4-t^2)dt$$$$\mathrm{I}\:=\:\int\:{t}\:\left(−\mathrm{4}{t}\left(\mathrm{4}−{t}^{\mathrm{2}} \right)\right)\:{dt}\:…

Question Number 99173 by bemath last updated on 19/Jun/20 Answered by MJS last updated on 19/Jun/20 $$\mathrm{this}\mathrm{is}\mathrm{very}\mathrm{complicated}$$$$\int \sqrt{\frac{1+\cot x}{\csc x+\cot x}}dx=\int \sqrt{\frac{\sin x+\cos x}{1+\cos x}}dx=$$$$[t=\tan \frac{x}{2}\to dx={2\cos }^2\frac{x}{2}dt]$$$$=\sqrt{\mathrm{2}}\int\frac{\sqrt{−{t}^{\mathrm{2}} +\mathrm{2}{t}+\mathrm{1}}}{{t}^{\mathrm{2}}…

Question Number 33619 by NECx last updated on 20/Apr/18 $$\int x^{5/2}{(1-x)}^{3/2}dx$$ Commented by abdo imad last updated on 20/Apr/18 $${let}\:{put}\:{I}\:=\:\int\:\:{x}^{\frac{\mathrm{5}}{\mathrm{2}}} \:\left(\mathrm{1}−{x}\right)^{\frac{\mathrm{3}}{\mathrm{2}}} \:{dx}\:\:.{ch}.{x}={sin}^{\mathrm{2}}…

Question Number 164685 by Zaynal last updated on 20/Jan/22 $$\int 4{\mathit{x}}^2{(1-\mathit{x})}^3\mathit{d}\mathit{x}$$ Answered by Ar Brandon last updated on 20/Jan/22 $$=\mathrm{4}\int\left(\mathrm{1}−{x}−\mathrm{1}\right)^{\mathrm{2}} \left(\mathrm{1}−{x}\right)^{\mathrm{3}} {dx}…