Question

 ` ∫   1 /{x^{1/3} ( x^{1/3} -1)}   ` dx

Answer 1

  `  Let  I =  ∫   1 /{x^{1/3} ( x^{1/3} -1)}   ` dx

 ` ∫   1 / {x^{2/3} -   x^{1/3}}   ` dx
\[ \text{Let x }= t^3 \]
\[ \text{On differentiating both sides, we get}\]
\[ dx = 3 t^2 \text{  dt }\]
` ∴  I = ƒ {3 t^2}/ {( t^3 )^\{2/3} - ( t^3)^{1/3}}   dt  `

\[ = \int\frac{3 t^2}{t^2 - t}dt\]
\[ = 3\int\frac{t}{t - 1}dt\]
\[ = 3\int\frac{\left( t - 1 \right) + 1}{t - 1}dt\]
\[ = 3\int\left[ \left( 1 \right) + \frac{1}{t - 1} \right]dt\]
\[ = 3\left[ t + \text{ log }\left( t - 1 \right) \right] + c\]
` =  3 x ^{1/3 } + 3 log ( x^{1/3}  -1 ) + c `
Hence, ` ∫   1 /{x^{1/3} ( x^{1/3} -1)} dx = 3x^{1/3}   + 3  log  ( x^{1/3} -1 ) + c  `