Integrate the function: (x^ 3 -1 )^ 1 3 x^ 5 - Vidyadip

Question

Integrate the function: $\left(x^{3}-1\right)^{\frac{1}{3}} x^{5}$

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Answer

Let x³ - 1 = t
$\Rightarrow$ 3x²dx = dt
$\Rightarrow\int\left(x^{3}-1\right)^{\frac{1}{3}} x^{5} d x=\int\left(x^{3}-1\right)^{\frac{1}{3}} x^{3} \cdot x^{2} d x$
$= \int t^{\frac{1}{3}}(t+1) \frac{d t}{3}$
$=\frac{1}{3} \int\left(t^{\frac{4}{3}}+t^{\frac{1}{3}}\right) d t$
$=\frac{1}{3}\left[\frac{t^{\frac{7}{3}}}{\frac{7}{3}}+\frac{t^{\frac{4}{3}}}{\frac{4}{3}}\right]+C$
$= \frac{1}{3}\left[\frac{3}{7} t^{\frac{7}{3}}+\frac{3}{4} t^{\frac{4}{3}}\right]+C$
$= \frac{1}{7}\left(x^{3}-1\right)^{\frac{7}{3}}+\frac{1}{4}\left(x^{3}-1\right)^{\frac{4}{3}}+C$

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