Q223 / 270 · NDA 2017 (Phase 1) · Mathematics

What is $\int \frac{\mathrm{dx}}{\mathrm{x}\left(\mathrm{x}^7+1\right)}$ equal to?

  • A
    $\frac{1}{2} \ln \left|\frac{\mathrm{x}^7-1}{\mathrm{x}^7+1}\right|+\mathrm{c}$
  • B
    $\frac{1}{7} \ln \left|\frac{\mathrm{x}^7+1}{\mathrm{x}^7}\right|+\mathrm{c}$
  • C
    $\quad \ln \left|\frac{\mathrm{x}^7-1}{7 \mathrm{x}}\right|+\mathrm{c}$
  • Correct answer
    D
    $\frac{1}{7} \ln \left|\frac{\mathrm{x}^7}{\mathrm{x}^7+1}\right|+\mathrm{c}$

Solution

$\int \frac{\mathrm{dx}}{\mathrm{x}\left(\mathrm{x}^7+1\right)}=\int \frac{\mathrm{x}^6}{\mathrm{x}^7\left(\mathrm{x}^7+1\right)} \cdot \mathrm{dx}$ Let $\mathrm{x}^7=\mathrm{t}$ $\Rightarrow 7 x^6 . d x=\mathrm{dt} \Rightarrow x^6 d x=\frac{d t}{7}$ Then $\int \frac{x^6 d x}{x^7\left(x^7+1\right)}=\frac{1}{7} \int \frac{d t}{t(t+1)}$ $\begin{aligned} & =\frac{1}{7}\left[\int \frac{1}{t} \mathrm{dt}-\int \frac{1}{\mathrm{t}+1} \mathrm{dt}\right] \\ & =\frac{1}{7}[\ln |\mathrm{t}|-\ln |\mathrm{t}+1|]+\mathrm{c} \\ & =\frac{1}{7} \ln \left|\frac{\mathrm{t}}{\mathrm{t}+1}\right|+\mathrm{c} \\ & =\frac{1}{7} \ln \left|\frac{\mathrm{x}^7}{\mathrm{x}^7+1}\right|+\mathrm{c} \end{aligned}$

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

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