NCERT Class 12 · Ch 7 Exercise 7.2 · Q3
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.2 Question 3
Evaluate \(\int \frac{1}{x \log{\left(x \right)} + x}\, dx\).
💡 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: \(\log{\left(\left|{\log{\left(\left|{x}\right| \right)} + 1}\right| \right)}\) + C.
- Work on each open interval where the original integrand is real and defined.
- u=1+log x, du=dx/x; integrate du/u (x>0, x≠e^(−1)).
- An antiderivative is F(x)=\(\log{\left(\left|{\log{\left(\left|{x}\right| \right)} + 1}\right| \right)}\).
- Differentiate this expression to check that F′(x)=\(\frac{1}{x \log{\left(x \right)} + x}\). Add an arbitrary constant C on each domain component.
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