Category: Integration

Question Number 164682 by Zaynal last updated on 20/Jan/22 $$\int 27{\mathit{x}}^2\sqrt{3\mathit{x}-2}\mathit{d}\mathit{x}$$ Answered by Ar Brandon last updated on 20/Jan/22 $$=3\int {(3x-2+2)}^2\sqrt{3x-2}dx$$$$=\mathrm{3}\int\left(\left(\mathrm{3}{x}−\mathrm{2}\right)^{\frac{\mathrm{5}}{\mathrm{2}}}…

Question Number 33599 by abdo imad last updated on 19/Apr/18 $$calculatef\left(a\right)={\int }_{-a}^a\frac{dx}{{(t^2+x^2)}^{\frac{3}{2}}}witha>0.$$ Commented by abdo imad last updated on…

Question Number 33589 by abdo imad last updated on 19/Apr/18 $$1)decomposeF\left(x\right)=\frac{1}{(x^2+4){(x-3)}^2}$$$$2)calculate{\int }_4^{+\mathrm{\infty }}\frac{dx}{(x^2+4){(x-3)}^2}.$$ Commented by math khazana…

Question Number 99120 by  M±th+et+s last updated on 18/Jun/20 $$provethat:$$$${\int }_{-\frac{1}{2}}^{\mathrm{\infty }}{e}^{-(4x^6+12x^5+15x^4+10x^3+4x^2+x)}dx$$$$=\frac{\sqrt[{\mathrm{8}}]{{e}}}{\mathrm{3}}\left[\frac{\Gamma\left(\frac{\mathrm{1}}{\mathrm{6}}\right)^{\frac{−\mathrm{1}}{\mathrm{2}}} }{\mathrm{2}\sqrt[{\mathrm{3}}]{\mathrm{2}}}\mathrm{1}{F}\mathrm{2}\left(_{\mathrm{1}/\mathrm{3},\mathrm{2}/\mathrm{3}} ^{\mathrm{1}/\mathrm{6}} \mid\frac{−\mathrm{1}}{\mathrm{69}/\mathrm{2}}\:\right)\:+\frac{\Gamma\left(\mathrm{5}/\mathrm{6}\right)}{\mathrm{128}\sqrt[{\mathrm{3}}]{\mathrm{4}}}\mathrm{1}{F}\mathrm{2}\left(_{\mathrm{4}/\mathrm{3},\mathrm{5}/\mathrm{3}}…

Question Number 33570 by Joel578 last updated on 19/Apr/18 $$A=\underset{n=2}{\sum ^{2017}}[\underset{1}{\stackrel{n}{\int }}{2\tan }^{-1}x+{\sin }^{-1}\left(\frac{2x}{1+x^2}\right)dx]$$$${B}\:=\:\underset{{n}=\mathrm{2}} {\overset{\mathrm{2017}} {\prod}}\:\left[\underset{\mathrm{1}} {\overset{{n}} {\int}}\:\mathrm{2tan}^{−\mathrm{1}} \:{x}\:+\:\mathrm{sin}^{−\mathrm{1}} \left(\frac{\mathrm{2}{x}}{\mathrm{1}\:+\:{x}^{\mathrm{2}}…