Category: Integration

Question Number 142667 by mnjuly1970 last updated on 03/Jun/21 Answered by mindispower last updated on 03/Jun/21 $$T\left(n\right)={\int }_0^1{x}^{n}ln(1+x)dx=\frac{ln\left(2\right)}{n+1}-{\int }_0^1\frac{x^{n+1}}{n+1}.\frac{dx}{1+x}$$$$\mathrm{0}\leqslant\int_{\mathrm{0}}…

Question Number 142643 by ArielVyny last updated on 03/Jun/21 $${\int }_0^{\frac{\pi }{2}}\frac{{cos}^{2}t}{sint}dt$$ Answered by MJS_new last updated on 03/Jun/21 $$\int\frac{\mathrm{cos}^{\mathrm{2}} \:{t}}{\mathrm{sin}\:{t}}{dt}=\int\frac{\mathrm{1}−\mathrm{sin}^{\mathrm{2}} \:{t}}{\mathrm{sin}\:{t}}{dt}=\int\left(−\mathrm{sin}\:{t}\:+\mathrm{csc}\:{t}\right){dt}=…

Question Number 142629 by qaz last updated on 03/Jun/21 $$\underset{\mathrm{n}=1}{\sum ^{\mathrm{\infty }}}{(-1)}^{\mathrm{n}}\cdot \frac{2\mathrm{n}-1}{\left(2\mathrm{n}\right)!}\cdot {\left(\frac{\pi }{2}\right)}^{2\mathrm{n}}$$$$=\underset{\mathrm{n}=1}{\sum ^{\mathrm{\infty }}}\frac{2\mathrm{n}-1}{\left(2\mathrm{n}\right)!}\cdot {(-{\left(\frac{\pi }{2}\right)}^2)}^{\mathrm{n}}$$$$=\left(\mathrm{2xD}−\mathrm{1}\right)\mid_{\mathrm{x}=\pi/\mathrm{2}} \underset{\mathrm{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{x}^{\mathrm{2}} \right)^{\mathrm{n}} }{\left(\mathrm{2n}\right)!}…

Question Number 77089 by Dah Solu Tion last updated on 03/Jan/20 $$\mathit{C}\mathit{h}\mathit{e}\mathit{a}\mathit{p}\downdownarrows$$$$\int \sqrt{\frac{\mathit{x}}{\sqrt{\frac{\mathit{x}}{\sqrt{\frac{\mathit{x}}{\dots }}}}}}\mathit{d}\mathit{x}$$$$$$$$$$ Commented by Tony Lin last…

Question Number 11544 by Nayon last updated on 28/Mar/17 $$$$$$$$$$$$$$$$$$$$$$Evaluate\int x^3{e}^{2x}dx$$$$ \…