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Question Number 73483 by abdomathmax last updated on 13/Nov/19

$$find\int \frac{dx}{x+2-\sqrt{x^2-x+7}}$$

Answered by MJS last updated on 13/Nov/19

$$\int \frac{dx}{x+2-\sqrt{x^2-x+7}}=\int \frac{dx}{x+2-\sqrt{{(x-\frac{1}{2})}^2+\frac{27}{4}}}=$$$$[t=\frac{\sqrt{3}}{9}(2x-1)\to dx=\frac{3\sqrt{3}}{2}dt]$$$$=\int \frac{dt}{t+\frac{5\sqrt{3}}{9}-\sqrt{t^2+1}}=$$$$[t=\sinh \ln u=\frac{u^2-1}{2u}\Rightarrow u=t+\sqrt{t^2+1}\to dt=\frac{u^2+1}{2u^2}du]$$$$=\frac{3\sqrt{3}}{2}\int \frac{u^2+1}{u(5u-3\sqrt{3})}du=$$$$=\frac{3\sqrt{3}}{10}\int du-\frac{1}{2}\int \frac{du}{u}+\frac{26}{25}\int \frac{du}{u-\frac{3\sqrt{3}}{5}}=$$$$=\frac{3\sqrt{3}}{10}u-\frac{1}{2}\ln u+\frac{26}{25}\ln (5u-3\sqrt{3})=$$$$...$$$$=\frac{2x-1+2\sqrt{x^2-x+7}}{10}-\frac{1}{2}\ln (2x-1+2\sqrt{x^2-x+7})+\frac{26}{25}\ln (5x-16+5\sqrt{x^2-x+7})+C$$

Commented by abdomathmax last updated on 17/Nov/19

$$thankssirmjs.$$

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