Q110 / 160 · AP EAMCET 2024 (20 May Shift 1) · Mathematics

$\int \frac{3 x^9+7 x^8}{\left(x^2+2 x+5 x^8\right)^2} d x=$

  • A
    $\frac{x^7}{5 x^7+x+2}+c$
  • Correct answer
    B
    $\frac{x^7}{2\left(5 x^7+x+2\right)}+c$
  • C
    $\frac{1}{2\left(5 x^7+x+2\right)}+c$
  • D
    $\frac{-x^7}{2\left(5 x^7+x+2\right)}+c$

Solution

$I=\int \frac{3 x^9+7 x^8}{\left(x^2+2 x+5 x^8\right)^2} d x$ $I=\int \frac{3 x^9+7 x^8}{x^{16}\left(\frac{1}{x^6}+\frac{2}{x^7}+5\right)^2} d x \Rightarrow I=\int \frac{3 x^{-7}+7 x^{-8}}{\left(\frac{1}{x^6}+\frac{2}{x^7}+5\right)^2} d x$ Let $\frac{1}{x^6}+\frac{2}{x^7}+5=t \Rightarrow\left(-6 x^{-7}-14 x^{-8}\right) d x=d t$ $\therefore I=\frac{-1}{2} \int \frac{1}{t^2} d t \Rightarrow I=\frac{1}{2 t}+C$ $\Rightarrow I=\frac{1}{2\left(\frac{1}{x^6}+\frac{2}{x^7}+5\right)}+C \Rightarrow I=\frac{x^7}{2\left(x+2+5 x^7\right)}+C$

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

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