Q109 / 160 · AP EAMCET 2024 (20 May Shift 2) · Mathematics

$\int \frac{d x}{x\left(x^4+1\right)}=$

  • A
    $\log \left(\frac{x}{x^4+1}\right)+c$
  • B
    $\frac{3}{4} \log \left(x^4+1\right)+c$
  • C
    $\frac{1}{3} \log \left(\frac{x^3}{x^4+1}\right)+c$
  • Correct answer
    D
    $\frac{1}{4} \log \left(\frac{x^4}{x^4+1}\right)+c$

Solution

$\mathrm{I}=\int \frac{d x \times x^3}{x\left(x^4+1\right) \times x^3} \Rightarrow \mathrm{I}=\int \frac{x^3 d x}{x^4\left(x^4+1\right)} d x$ $\begin{aligned} & \text { Let } x^4=t \Rightarrow 4 x^3=d t \\ & \therefore \mathrm{I}=\frac{1}{4} \int \frac{d t}{t(t+1)}=\frac{1}{4} \int\left[\frac{1}{t}-\frac{1}{t+1}\right] d t \\ & =\frac{1}{4}[\log (t)-\log (t+1)]+\mathrm{C}=\frac{1}{4} \log \left(\frac{t}{t+1}\right) \\ & \mathrm{I}=\frac{1}{4} \log \left(\frac{x^4}{x^4+1}\right)+\mathrm{C}\end{aligned}$

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

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