Q113 / 160 · AP EAMCET 2023 (16 May Shift 2) · Mathematics

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

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

Solution

Let $I=\int \frac{\sqrt{x^4+x^{-4}+2}}{x^3} d x$ $\begin{aligned} & \Rightarrow I=\int \frac{\sqrt{\left(x^2\right)^2+\left(x^{-2}\right)^2+2 x^2 x^{-2}}}{x^3} d x \\ & =\int\left(\frac{x^2+x^{-2}}{x^3}\right) d x=\int\left(\frac{1}{x}+\frac{1}{x^5}\right) d x \\ & =\ln |x|-\frac{1}{4 x^4}+C\end{aligned}$

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

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