Category: Integration

Question Number 128841 by mnjuly1970 last updated on 10/Jan/21 $$\dots nicecalculus\dots$$$$\mathcal{E}valuationof::\mathrm{\Phi }={\int }_0^{1}ln\left(x\right).arctan\left(x\right)dx$$$$solution::$$$$note1::\underset{n=1}{\sum ^{\mathrm{\infty }}}\frac{{(-1)}^{n-1}}{2n-1}=arctan\left(1\right)=\frac{\pi }{4}$$$${note}\:\mathrm{2}\:::\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}−\mathrm{1}}…

Question Number 128826 by benjo_mathlover last updated on 10/Jan/21 $${\int }_{-1}^5\sqrt{\frac{{2\mathrm{x}}^2-8}{\mathrm{x}}}\mathrm{dx}=?$$ Answered by bobhans last updated on 10/Jan/21 $$\:\int\:\sqrt{\frac{\left({x}\sqrt{\mathrm{2}}\right)^{\mathrm{2}} −\left(\sqrt{\mathrm{8}}\right)^{\mathrm{2}} }{{x}}}\:{dx}\:…

Question Number 63268 by aliesam last updated on 01/Jul/19 Terms of Service Privacy Policy Contact: info@tinkutara.com

Question Number 63261 by aliesam last updated on 01/Jul/19 $$\int xtan\left(x\right)dx$$ Commented by kaivan.ahmadi last updated on 01/Jul/19 $$u=tanx\Rightarrow du=\frac{dx}{1+x^2}$$$${dv}={xdx}\Rightarrow{v}=\frac{{x}^{\mathrm{2}} }{\mathrm{2}} \…

Question Number 128775 by bramlexs22 last updated on 10/Jan/21 $$\int \frac{\mathrm{dx}}{{(1-\mathrm{x})}^2\sqrt{1-{\mathrm{x}}^2}}?$$ Answered by liberty last updated on 10/Jan/21 $$\:\mathrm{let}\:\varphi\:=\:\sqrt{\frac{\mathrm{1}+\mathrm{x}}{\mathrm{1}−\mathrm{x}}}\:\Rightarrow\mathrm{x}\:=\:\frac{\varphi^{\mathrm{2}} −\mathrm{1}}{\varphi^{\mathrm{2}} +\mathrm{1}}\:\mathrm{and}\:\mathrm{dx}\:=\:\frac{\mathrm{4}\varphi}{\left(\varphi^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{2}}…