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NDA 2017
Exam
NDA
Year
2017
Subject
Mathematics
Chapter
Indefinite Integration

NDA 2017 · Mathematics · Indefinite Integration

What is ∫ dx x(x^7+1) equal to?

What is ∫dxx(x7+1)\int \frac{\mathrm{dx}}{\mathrm{x}\left(\mathrm{x}^7+1\right)} equal to?

Options

  1. A12ln⁡∣x7−1x7+1∣+c\frac{1}{2} \ln \left|\frac{\mathrm{x}^7-1}{\mathrm{x}^7+1}\right|+\mathrm{c}
  2. B17ln⁡∣x7+1x7∣+c\frac{1}{7} \ln \left|\frac{\mathrm{x}^7+1}{\mathrm{x}^7}\right|+\mathrm{c}
  3. Cln⁡∣x7−17x∣+c\quad \ln \left|\frac{\mathrm{x}^7-1}{7 \mathrm{x}}\right|+\mathrm{c}
  4. D17ln⁡∣x7x7+1∣+c\frac{1}{7} \ln \left|\frac{\mathrm{x}^7}{\mathrm{x}^7+1}\right|+\mathrm{c}

Correct answer

D. 17ln⁡∣x7x7+1∣+c\frac{1}{7} \ln \left|\frac{\mathrm{x}^7}{\mathrm{x}^7+1}\right|+\mathrm{c}

Step-by-step solution

∫dxx(x7+1)=∫x6x7(x7+1)⋅dx\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 x7=t⇒7x6.dx=dt⇒x6dx=dt7\mathrm{x}^7=\mathrm{t}\Rightarrow 7 x^6. d x=\mathrm{dt} \Rightarrow x^6 d x=\frac{d t}{7} Then ∫x6dxx7(x7+1)=17∫dtt(t+1)\int \frac{x^6 d x}{x^7\left(x^7+1\right)}=\frac{1}{7} \int \frac{d t}{t(t+1)}=17[∫1tdt−∫1t+1dt]=17[ln⁡∣t∣−ln⁡∣t+1∣]+c=17ln⁡∣tt+1∣+c=17ln⁡∣x7x7+1∣+c\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}$$
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