Category: Integration

Question Number 120599 by MagdiRagheb last updated on 01/Nov/20 $$Letx=u^{6}dx=6u^{5}du$$$$I=\int \frac{u^3}{{(1+u^2)}^2}\times 6u^{5}du$$$$=6\int \frac{u^8}{1+2u^2+u^4}du$$$$\:\:\:=\mathrm{6}\:\int\left[\frac{−\mathrm{4}}{{u}^{\mathrm{2}}…

Question Number 120582 by MagdiRagheb last updated on 01/Nov/20 $$Letu=x^{\frac{3}{5}}du=\frac{3}{5x^{\frac{2}{5}}}$$$$I=\frac{5}{3}\int \frac{u}{\sqrt{3-2u}}du=-\frac{5}{6}\int \frac{3-2u-3}{\sqrt{3-2u}}du$$$$=-\frac{5}{3}\int [\sqrt{3-2u}-3{(3-2u)}^{-\frac{1}{2}}]du$$$$=-\frac{5}{3}[-\frac{1}{3}{(3-2u)}^{\frac{3}{2}}+3{(3-2u)}^{\frac{1}{2}}]+c$$$$=\:−\frac{\mathrm{5}}{\mathrm{18}}\left(\mathrm{3}−\mathrm{2}{u}\right)^{\frac{\mathrm{1}}{\mathrm{2}}} \left[−\mathrm{3}+\mathrm{2}{u}\:+\:\mathrm{9}\right]+{c} \…

Question Number 120562 by peter frank last updated on 01/Nov/20 Answered by Dwaipayan Shikari last updated on 01/Nov/20 $$\int 1-\frac{bx+12}{x^2+bx+12}dx$$$$=x-b\int \frac{x+\frac{12}{b}}{x^2+bx+12}dx$$$$={x}−\frac{{b}}{\mathrm{2}}\int\frac{\mathrm{2}{x}+{b}}{{x}^{\mathrm{2}}…

Question Number 120552 by talminator2856791 last updated on 01/Nov/20 $$$$$$$$$$\mathrm{evaluate}:{\int }_0^{\mathrm{\infty }}{\left(\frac{x}{x+1}\right)}^{x!}dx$$$$$$$$$$ Terms of…

Question Number 120549 by bobhans last updated on 01/Nov/20 Answered by TANMAY PANACEA last updated on 01/Nov/20 $$\frac{-1}{4}\int \frac{-4x^2+4x+3-4x-3}{{(3+4x-4x^2)}^{\frac{3}{2}}}dx$$$$=\frac{−\mathrm{1}}{\mathrm{4}}\int\frac{{dx}}{\:\sqrt{\mathrm{3}+\mathrm{4}{x}−\mathrm{4}{x}^{\mathrm{2}} }}+\frac{\mathrm{1}}{\mathrm{4}}\int\frac{\mathrm{4}{x}+\mathrm{3}}{\left(\mathrm{3}+\mathrm{4}{x}−\mathrm{4}{x}^{\mathrm{2}} \right)^{\frac{\mathrm{3}}{\mathrm{2}}}…