Q72 / 170 · Manipal (MET) 2016 · Mathematics

$\int \frac{\left(2 x^{12}+5 x^9\right)}{\left(1+x^3+x^5\right)^3} d x$ equals

  • Correct answer
    A
    $\frac{x^{10}}{2\left(x^5+x^3+1\right)^2}+C$
  • B
    $\frac{x^2+2 x}{\left(x^5+x^3+1\right)}+C$
  • C
    $\left.\log \left(x^5+x^3+1+\sqrt{\left(2 x^{12}+5 x^9\right)}+C\right)\right]$
  • D
    None of the above

Solution

Let $I=\int \frac{2 x^{12}+5 x^9}{\left(1+x^3+x^5\right)^3} d x=\int \frac{5 x^{-6}+2 x^{-3}}{\left(1+x^{-2}+x^{-5}\right)^3} d x$ Put $1+x^{-2}+x^{-5}=t,\left(-2 x^{-3}-5 x^{-6}\right) d x=d t$ $\therefore \quad I=-\int \frac{d t}{t^3}=\frac{1}{2 t^2}=\frac{1}{2\left(1+x^{-2}+x^{-5}\right)^2}$ $=\frac{x^{10}}{2\left(x^5+x^3+1\right)^2}+C$

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

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