Q111 / 160 · AP EAMCET 2022 (07 Jul Shift 2) · Mathematics

$\int \frac{x^8+4}{x^4-2 x^2+2} d x=A x^5+B x^3+C x+k$, then $5 A+3 B+C=$

  • A
    $7$
  • Correct answer
    B
    $5$
  • C
    $3$
  • D
    $1$

Solution

$\begin{aligned} & \int \frac{x^8+4}{x^4-2 x^2+2} d x=\int \frac{x^8+4 x^4+4-4 x^4}{x^4-2 x^2+2} d x \\ & =\int \frac{\left(x^4+2\right)^2-\left(2 x^2\right)^2}{x^4-2 x^2+2} d x=\int x^4+2+2 x^2 d x \\ & =\frac{x^5}{5}+\frac{2 x^3}{3}+2 x+k \\ \Rightarrow & A=\frac{1}{5}, B=\frac{2}{3}, C=2 \\ & 5 A+3 B+C=5\end{aligned}$

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

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