Evaluate the following : x^3 x-1 d x - Vidyadip

Question

Evaluate the following : $\int \frac{x^3}{x-1} \cdot d x$

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Answer

$
\begin{aligned}
\text { I } & =\int \frac{x^3-1+1}{x-1} \cdot d x \\
& =\int\left(\frac{x^3-1}{x-1}+\frac{1}{x-1}\right) \cdot d x \\
& =\int\left(\frac{(x-1)\left(x^2+x+1\right)}{(x-1)}+\frac{1}{x-1}\right) \cdot d x \\
& =\int\left(x^2+x+1+\frac{1}{x-1}\right) \cdot d x \\
& =\frac{x^3}{3}+\frac{x^2}{2}+x+\log (x-1)+c
\end{aligned}
$

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