Mathematics

Evaluate the following integrals:
$$\displaystyle \int { \cfrac { x }{ \left( { x }^{ 2 }+4 \right) \sqrt { { x }^{ 2 }+9 }  }  } dx\quad $$


SOLUTION
Now,
$$\displaystyle\int { \cfrac { x }{ \left( { x }^{ 2 }+4 \right) \sqrt { { x }^{ 2 }+9 }  }  } dx\quad $$

Put $$x^2+9=p^2$$......(1).

or, $$2x\ dx=2p\ dp$$.

Now using these in the above expression we get,

$$=\displaystyle\int { \cfrac { p }{\left( { p }^{ 2 }+5 \right)  p }  } dp\quad $$

$$=\displaystyle\int { \cfrac { 1 }{ \left( { p }^{ 2 }+5 \right)   }  } dp\quad $$

$$=\displaystyle\int { \cfrac { 1 }{ \left( { p }^{ 2 }+(\sqrt{5})^2 \right)   }  } dp\quad $$

$$=\dfrac{1}{2\sqrt{5}}\log\left|\dfrac{p-\sqrt{5}}{p+\sqrt{5}}\right|+c$$

$$=\dfrac{1}{2\sqrt{5}}\log\left|\dfrac{\sqrt{x^2+9}-\sqrt{5}}{\sqrt{x^2+9}+\sqrt{5}}\right|+c$$ [ Using (1)]
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Subjective Medium Published on 17th 09, 2020
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