NCERT Class 12 · Ch 7 Miscellaneous Exercise · Q1
NCERT Class 12 Maths · Chapter 7 Integrals · Miscellaneous Exercise Question 1
Evaluate \(\int \frac{1}{- x^{3} + x}\, dx\).
💡 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: \(\log{\left(\left|{x}\right| \right)} - \frac{\log{\left(\left|{x^{2} - 1}\right| \right)}}{2}\) + C.
- Work on each open interval where the original integrand is real and defined.
- Partial fractions give 1/x+x/(1−x²); integrate both logarithmic derivatives.
- An antiderivative is F(x)=\(\log{\left(\left|{x}\right| \right)} - \frac{\log{\left(\left|{x^{2} - 1}\right| \right)}}{2}\).
- Differentiate this expression to check that F′(x)=\(\frac{1}{- x^{3} + x}\). Add an arbitrary constant C on each domain component.
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