NCERT Class 12 · Ch 7 Exercise 7.2 · Q14
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.2 Question 14
Evaluate \(\int \frac{\log{\left(x \right)}^{- m}}{x}\, dx\). Here x>0 and m≠1; retain the real-power domain restrictions.
💡 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 x)^(1−m)/(1−m)+C, m≠1, on each interval where the original real power is defined.
- Work on each open interval where the original integrand is real and defined.
- u=log x, du=dx/x; integrate u^(−m)du with m≠1. Use real-power domain components (always x>1 for arbitrary real m).
- An antiderivative is F(x)=\(\frac{\log{\left(\left|{x}\right| \right)}^{1 - m}}{1 - m}\).
- Differentiate this expression to check that F′(x)=\(\frac{\log{\left(x \right)}^{- m}}{x}\). Add an arbitrary constant C on each domain component.
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