NCERT Class 12 · Ch 7 Exercise 7.5 · Q16
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.5 Question 16
Evaluate \(\int \frac{1}{x \left(x^{n} + 1\right)}\, dx\).
💡 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: For n≠0: log|x|−log|1+xⁿ|/n+C wherever the original real power is defined. For n=0,x≠0: (1/2)log|x|+C.
- Work for x>0 and real n≠0; extend to other real-power components only where the original power is defined.
- For n≠0 put u=xⁿ, giving dx/x=du/(nu). Split 1/[u(u+1)]=1/u−1/(u+1).
- If n=0, the integrand is 1/(2x), whose primitive is (1/2)log|x|.
- An antiderivative is F(x)=\(\log{\left(\left|{x}\right| \right)} - \frac{\log{\left(\left|{x^{n} + 1}\right| \right)}}{n}\).
- Differentiate this expression to check that F′(x)=\(\frac{1}{x \left(x^{n} + 1\right)}\). Add an arbitrary constant C on each domain component.
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