Q114 / 160 · AP EAMCET 2021 (24 Aug Shift 1) · Mathematics

$\int \frac{x^3-x^2+x-1}{x-1} d x$ is equal to

  • A
    $\frac{x^3}{3}-x+c$
  • B
    $\frac{x^2}{3}+x+c$
  • Correct answer
    C
    $\frac{x^3}{3}+x+c$
  • D
    $2 x+c$

Solution

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

Hence, the correct answer is option: 'C'.

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