NCERT Class 12 · Ch 7 Exercise 7.2 · Q36
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.2 Question 36
Evaluate \(\int \frac{\left(x + 1\right) \left(x + \log{\left(x \right)}\right)^{2}}{x}\, dx\).
💡 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: \(\frac{\left(x + \log{\left(\left|{x}\right| \right)}\right)^{3}}{3}\) + C.
- Work on each open interval where the original integrand is real and defined.
- u=x+log x, du=(x+1)dx/x, x>0.
- An antiderivative is F(x)=\(\frac{\left(x + \log{\left(\left|{x}\right| \right)}\right)^{3}}{3}\).
- Differentiate this expression to check that F′(x)=\(\frac{\left(x + 1\right) \left(x + \log{\left(x \right)}\right)^{2}}{x}\). Add an arbitrary constant C on each domain component.
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