Mathematics

$$\displaystyle \int \displaystyle \frac{xdx}{\sqrt{1+x^2+\sqrt{(1+x^2)^3}}}$$ is equal to


ANSWER

$$2\sqrt{1+\sqrt{1+x^2}+c}$$


SOLUTION
Let $$\displaystyle I=\int  \frac { xdx }{ \sqrt { 1+x^{ 2 }+\sqrt { (1+x^{ 2 })^{ 3 } }  }  } $$
Put $$1+{ x }^{ 2 }=t\Rightarrow 2xdx=dt$$

$$\displaystyle I=\frac { 1 }{ 2 } \int { \frac { dt }{ \sqrt { t+{ t }^{ 3 } }  }  } $$
Put $$\displaystyle u=\sqrt { t } \Rightarrow du=\frac { 1 }{ 2\sqrt { t }  } dt$$

$$\displaystyle I=\int { \frac { 1 }{ \sqrt { u+1 }  } du } =2\sqrt { u+1 } \\ =2\sqrt { \sqrt { t } +1 } =2\sqrt { \sqrt { 1+{ x }^{ 2 } } +1 } $$
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Single Correct Medium Published on 17th 09, 2020
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