Q113 / 160 · TS EAMCET 2019 (04 May Shift 2) · Mathematics

$\int \frac{2 x^3-4 x^2-x-3}{x^2-2 x-3} d x=$

  • A
    $\frac{7}{2} \log |x-1|+\frac{3}{2} \log |x+3|+c$
  • B
    $2 \log |x-1|+\frac{7}{2} \log |x+3|+c$
  • C
    $2 x+\frac{1}{2} \log |x+1|+\frac{3}{4} \log |x-3|+c$
  • Correct answer
    D
    $x^2+2 \log |x+1|+3 \log |x-3|+c$

Solution

We have, $$ \begin{aligned} & I=\int \frac{2 x^3-4 x^2-x-3}{x^2-2 x-3} d x \\ & I=\int\left(2 x+\frac{5 x-3}{(x+1)(x-3)}\right) d x \\ & I=\int\left(2 x+\frac{2}{x+1}+\frac{3}{x-3}\right) d x \\ & I=x^2+2 \log |x+1|+3 \log |x-3|+c \end{aligned} $$

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

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