NCERT Class 12 · Ch 7 Miscellaneous Exercise · Q8
NCERT Class 12 Maths · Chapter 7 Integrals · Miscellaneous Exercise Question 8
Evaluate \(\int \frac{x^{5} - x^{4}}{x^{3} - x^{2}}\, dx\). Original logarithm domain: x>0, x≠1.
💡 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: \(\frac{x^{3}}{3}\) + C.
- Work on each open interval where the original integrand is real and defined.
- On x>0,x≠1 the integrand simplifies to (x⁵−x⁴)/(x³−x²)=x²; retain the original hole at 1.
- An antiderivative is F(x)=\(\frac{x^{3}}{3}\).
- Differentiate this expression to check that F′(x)=\(\frac{x^{5} - x^{4}}{x^{3} - x^{2}}\). Add an arbitrary constant C on each domain component.
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