Question

Evaluate $\int (x^3-1{)}^{\frac{1}{3}}.x^{5}dx$

Answer 1

$\int {\left(x^3-1\right)}^{\frac{1}{3}}{x}^{5}dx$

$=\int {\left(x^3-1\right)}^{\frac{1}{3}}{x}^5{x}^{2}dx$

Putting $x^3-1=t$

$3x^{2}dx=dt$ $\Rightarrow x^{2}dx=\frac{dt}{3}$

$=\frac{1}{3}\int t^{\frac{1}{3}}\left(t+1\right)dt$

$=\frac{1}{3}\int \left(t^{\frac{4}{3}}+t^{\frac{1}{3}}\right)dt$

$=\frac{1}{3}\left[\frac{3}{7}{t}^{\frac{7}{3}}+\frac{3}{4}{t}^{\frac{4}{3}}\right]+c$

$=\frac{1}{7}{\left(x^3-1\right)}^{\frac{7}{3}}+\frac{1}{4}{\left(x^3-1\right)}^{\frac{4}{3}}+c$