Question and Answers Forum

All Questions      Topic List

Integration Questions

Previous in All Question      Next in All Question      

Previous in Integration      Next in Integration      

Question Number 64419 by turbo msup by abdo last updated on 17/Jul/19

$$1)find\int \frac{dx}{x-\sqrt{1-x^2}}$$$$2)calculate{\int }_0^1\frac{dx}{x-\sqrt{1-x^2}}$$

Commented by mathmax by abdo last updated on 18/Jul/19

$$1)letA=\int \frac{dx}{x-\sqrt{1-x^2}}changementx=sin\theta give$$$$A=\int \frac{cos\theta }{sin\theta -cos\theta }d\theta =\int \frac{cos\theta }{cos\theta (tan\theta -1)}d\theta$$$$=\int \frac{d\theta }{tan\theta -1}{=}_{tan\theta =t}\int \frac{dt}{(1+t^2)(t-1)}letdecompose$$$$F\left(t\right)=\frac{1}{(t-1)(t^2+1)}\Rightarrow F\left(t\right)=\frac{a}{t-1}+\frac{bt+c}{t^2+1}$$$$a={lim}_{t\to 1}(t-1)F\left(t\right)=\frac{1}{2}$$$${lim}_{t\to +\mathrm{\infty }}tF\left(t\right)=0=a+b\Rightarrow b=-\frac{1}{2}\Rightarrow F\left(t\right)=\frac{1}{2(t-1)}+\frac{-\frac{1}{2}t+c}{t^2+1}$$$$F\left(0\right)=-1=-\frac{1}{2}+c\Rightarrow c=-\frac{1}{2}\Rightarrow F\left(t\right)=\frac{1}{2(t-1)}-\frac{1}{2}\frac{t+1}{t^2+1}\Rightarrow$$$$A=\frac{1}{2}\int \frac{dt}{t-1}-\frac{1}{2}\int \frac{t+1}{t^2+1}dt+c$$$$=\frac{1}{2}ln\mid t-1\mid -\frac{1}{4}ln(t^2+1)-\frac{1}{2}arctan\left(t\right)+c$$$$wehavet=tan\theta and\theta =arcsinx\Rightarrow$$$$A=\frac{1}{2}ln\mid tan\left(arcsinx\right)-1\mid -\frac{1}{4}ln(1+{tan}^2\left(arcsinx\right))-\frac{1}{2}arcsinx+c$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com