Q60 / 150 · MHT CET 2012 · Mathematics

The value of $\int \frac{d x}{x\left(x^{n}+1\right)}$ is

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

Solution

Let $\begin{aligned} I=\int \frac{d x}{x\left(x^{n}+1\right)} \\ \left.\quad \text { (let } t=x^{n}+1, d t=n x^{n-1} d x\right) \\=& \int \frac{d t}{r x^{n} \cdot t} \quad\left(\frac{d t}{n x^{n}}=\frac{d x}{x}\right) \\=& \frac{1}{n} \int \frac{d t}{t(t-1)} \\=& \frac{1}{n} \int\left\{\frac{1}{t-1}-\frac{1}{t}\right\} d t \\=& \frac{1}{n}\{\log (t-1)-\log t\}+C \\=& \frac{1}{n} \log \frac{t-1}{t}+C \\=& \frac{1}{n} \log \frac{x^{n}}{x^{n}+1}+C \end{aligned}$

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

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