Mathematics

Evaluate the following integral:
$$\displaystyle \int { \cfrac { 1 }{ \sqrt { { (2-x) }^{ 2 }-1 }  }  } dx\quad $$


SOLUTION
$$\displaystyle\int{\dfrac{dx}{\sqrt{{\left(2-x\right)}^{2}-1}}}$$

Let $$t=2-x$$

$$\Rightarrow\,dt=-dx$$

$$=-\displaystyle\int{\dfrac{dt}{\sqrt{{t}^{2}-1}}}$$

We know that $$\displaystyle\int{\dfrac{dx}{\sqrt{{x}^{2}-{a}^{2}}}}=\log{\left|x+\sqrt{{x}^{2}-{a}^{2}}\right|}+c$$

Replace $$x\rightarrow\,t$$ and $$a\rightarrow\,1$$

$$=-\log{\left|t+\sqrt{{t}^{2}-1}\right|}+c$$

$$=-\log{\left|2-x+\sqrt{{\left(2-x\right)}^{2}-1}\right|}+c$$ where $$t=2-x$$

$$=-\log{\left|2-x+\sqrt{4+{x}^{2}-2x-1}\right|}+c$$

$$=-\log{\left|2-x+\sqrt{{x}^{2}-2x+3}\right|}+c$$

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Subjective Medium Published on 17th 09, 2020
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