Mathematics

Evaluate the following integral:
$$\displaystyle \int { \cfrac { 1 }{ \sqrt { { a }^{ 2 }+{ b }^{ 2 }{ x }^{ 2 } }  }  } dx$$


SOLUTION
$$\displaystyle\int{\dfrac{dx}{\sqrt{{a}^{2}+{b}^{2}{x}^{2}}}}$$

$$=\displaystyle\int{\dfrac{dx}{\sqrt{{a}^{2}+{\left(bx\right)}^{2}}}}$$

Let $$t=bx\Rightarrow\,dt=b\,dx$$

$$=\dfrac{1}{b}\displaystyle\int{\dfrac{dx}{\sqrt{{a}^{2}+{t}^{2}}}}$$

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

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

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

$$=\dfrac{1}{b}\log{\left|bx+\sqrt{{b}^{2}{x}^{2}+{a}^{2}}\right|}+c$$ where $$t=bx$$
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Subjective Medium Published on 17th 09, 2020
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