Q97 / 180 · COMEDK 2013 · Mathematics

$\int \frac{d x}{x\left(x^{7}+1\right)}$ is equal to

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

Solution

Let $\begin{aligned} I &=\int \frac{1}{x\left(x^{7}+1\right)} d x \\ &=\int \frac{x^{6}}{x^{7}\left(x^{7}+1\right)} d x \end{aligned}$ Put $x^{7}=t \Rightarrow 7 x^{6} d x=d t$ $$ \begin{aligned} \therefore \quad I &=\frac{1}{7} \int \frac{d}{t(t+1)}=\frac{1}{7} \int\left(\frac{1}{t}-\frac{1}{t+1}\right) d t \\ &=\frac{1}{7}[\log t-\log (t+1)]+c \\ &=\frac{1}{7} \log \left(\frac{t}{t+1}\right)+c=\frac{1}{7} \log \left(\frac{x^{7}}{x^{7}+1}\right)+c \end{aligned} $$

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

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