Category: Integration

Question Number 20157 by ajfour last updated on 23/Aug/17 Commented by ajfour last updated on 23/Aug/17 $$Computethevolumecommontothe$$$$sphere{x}^2+y^2+z^2=4andtheinsideof$$$${the}\:{paraboloid}\:\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}}…

Question Number 151221 by peter frank last updated on 19/Aug/21 $$\int \frac{\sin \mathrm{x}}{\sin (\mathrm{x}-\mathrm{a})}\mathrm{dx}$$ Commented by puissant last updated on 19/Aug/21 $$Q=\int \frac{sinx}{sin(x-a)}dx$$$$=\int \frac{sin(x-a+a)}{sin(x-a)}dx$$$$=\int\frac{{sin}\left({x}−{a}\right){cosa}+{cos}\left({x}−{a}\right){sina}}{{sin}\left({x}−{a}\right)}{dx}…

Question Number 85669 by john santu last updated on 23/Mar/20 $$\int \frac{\sqrt{1+x}}{\sqrt{1-x}}dx$$$$$$ Commented by mathmax by abdo last updated on 24/Mar/20 $${I}\:=\int\:\frac{\sqrt{\mathrm{1}+{x}}}{\:\sqrt{\mathrm{1}−{x}}}{dx}\:\Rightarrow\:{I}\:=\int\sqrt{\frac{\mathrm{1}+{x}}{\mathrm{1}−{x}}}{dx}\:{we}\:{use}\:{the}\:{changement}\:…

Question Number 85648 by sakeefhasan05@gmail.com last updated on 23/Mar/20 $$\int \frac{({\mathrm{x}}^3-4)}{(\mathrm{x}+1)}\mathrm{dx}$$ Answered by petrochengula last updated on 23/Mar/20 $$=\int \frac{x^3+1-5}{(x+1)}dx$$$$=\int\frac{{x}^{\mathrm{3}} +\mathrm{1}}{{x}+\mathrm{1}}{dx}−\mathrm{5}\int\frac{\mathrm{1}}{{x}+\mathrm{1}}{dx}…

Question Number 85646 by M±th+et£s last updated on 23/Mar/20 $$showthat$$$$\int \frac{1}{{[x(x-1)(x-2)(x-3)\dots (x-m)]}^2}dx=$$$$=\frac{\mathrm{1}}{\left({m}!\right)^{\mathrm{2}} }\underset{{n}=\mathrm{0}} {\overset{{m}} {\sum}}\frac{$$\left(\begin{array}{c}m\\ n\end{array}\right)$$^{\mathrm{2}} }{{n}−{x}}+\frac{\mathrm{2}}{\left({m}!\right)^{\mathrm{2}} }{ln}\mid\underset{{n}=\mathrm{0}} {\overset{{m}} {\prod}}\left({x}−{n}\right)^{$$\left(\begin{array}{c}m\\ n\end{array}\right)$$^{\mathrm{2}} \left({H}_{{m}−{n}} −{H}_{{n}} \right)}…