NCERT Class 12 · Ch 7 Miscellaneous Exercise · Q17
NCERT Class 12 Maths · Chapter 7 Integrals · Miscellaneous Exercise Question 17
Integrate f′(ax+b)[f(ax+b)]ⁿ.
💡 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: For a≠0,n≠−1: [f(ax+b)]^(n+1)/[a(n+1)]+C. For n=−1: log|f(ax+b)|/a+C.
- Put u=f(ax+b); du=a f′(ax+b)dx, with a≠0.
- Integrate uⁿdu/a by the power rule when n≠−1, or du/(au) when n=−1.
- If a=0, the integrand is the constant f′(b)[f(b)]ⁿ where defined, so its primitive is x f′(b)[f(b)]ⁿ+C. Keep the original real-power domain restrictions.
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