Mathematics

Evaluate: $$\displaystyle \int \frac{dx}{x^{1/5}\left ( 1+x^{4/5} \right )^{1/2}}$$


ANSWER

$$\displaystyle \frac{5}{2}\sqrt{1+x^{4/5}}+k$$


SOLUTION
$$\displaystyle \int \frac{dx}{x^{1/5}\left ( 1+x^{4/5} \right )^{1/2}}$$

Put $$1+x^{ 4/5 }=t$$
$$\displaystyle \frac { 4 }{ 5 } { x }^{ -1/5 }dx=dt$$
$$\displaystyle \frac { dx }{ x^{ 1/5 } } =\frac { 5 }{ 4 } dt$$

So, 
$$I=\displaystyle \frac { 5 }{ 4 } \int  \frac { dt }{ t^{ 1/2 } } $$

$$\displaystyle I=\frac { 5 }{ 2 } t^{ 1/2 }+C$$

$$\displaystyle I=\frac { 5 }{ 2 } (1+x^{ 4/5 })^{ 1/2 }+C$$
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Single Correct Medium Published on 17th 09, 2020
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