NCERT Class 12 · Ch 7 Exercise 7.5 · Q20
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.5 Question 20
Evaluate \(\int \frac{1}{x \left(x^{4} - 1\right)}\, dx\).
💡 Concept Hints
- Identify the domain and the inner and outer functions, or the relevant one-sided limits.
- Apply the appropriate differentiation or limit rule, keep all chain factors, and check exceptional points.
✅ Step-by-Step Solution
Final Answer: \(- \log{\left(\left|{x}\right| \right)} + \frac{\log{\left(\left|{x^{4} - 1}\right| \right)}}{4}\) + C.
- Work on each open interval where the original integrand is real and defined.
- Polynomial division/partial fractions give \(\frac{x}{2 \left(x^{2} + 1\right)} + \frac{1}{4 \left(x + 1\right)} + \frac{1}{4 \left(x - 1\right)} - \frac{1}{x}\).
- Integrate each rational term using logarithms and ∫dx/(x²+A²)=tan⁻¹(x/A)/A.
- An antiderivative is F(x)=\(- \log{\left(\left|{x}\right| \right)} + \frac{\log{\left(\left|{x^{4} - 1}\right| \right)}}{4}\).
- Differentiate this expression to check that F′(x)=\(\frac{1}{x \left(x^{4} - 1\right)}\). Add an arbitrary constant C on each domain component.
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