Question

After Long.......


Do Disss Guyzzzz!!!

Btw New update is cool but I guess comments R still not posting​

Answer 1

Step-by-step explanation:

Given to integrate is

$\frac{ \sqrt[3]{x} }{x( \sqrt{x} + \sqrt[3]{x} )}$

Now we have,

$\int \frac{ \sqrt[3]{x} dx}{x( \sqrt{x} + \sqrt[3]{x} )}$

Let

$x = {y}^{6} \: = > dx = 6 {y}^{5} dy$

so,

$\int \frac{ {y}^{2} . \: 6 {y}^{5} \: dy }{ {y}^{6}( {y}^{3} + {y}^{2} ) }$

$= > \int \frac{6 {y}^{7} dy}{ {y}^{7}( {y}^{2} + y)}$

$= > 6 \int \frac{dy}{ {y}^{2} + y}$

$= > 6 \int \frac{dy}{ {y}^{2} + y + { (\frac{1}{2}) }^{2} - {( \frac{1}{2} })^{2} }$

$= > 6 \int \frac{dy}{(y + \frac{1}{2} )^{2} - {( \frac{1}{2} )}^{2} }$

$= > 6 \times \frac{1}{2 \times \frac{1}{2} } ln | \frac{(y + \frac{1}{2} ) - \frac{1}{2} }{(y + \frac{1}{2}) + \frac{1}{2} } | + c$

$= > 6 \: ln | \frac{y}{y + 1} | + c$

$= >6 \: ln | \frac{ \sqrt[6]{x} }{ \sqrt[6]{x} + 1} | + c$