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NCERT Class 12 · Ch 7 Exercise 7.5 · Q6

NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.5 Question 6

Evaluate \(\int \frac{1 - x^{2}}{x \left(1 - 2 x\right)}\, dx\).

💡 Concept Hints

  1. Identify the domain and the inner and outer functions, or the relevant one-sided limits.
  2. Apply the appropriate differentiation or limit rule, keep all chain factors, and check exceptional points.

✅ Step-by-Step Solution

Final Answer: \(\frac{x}{2} + \log{\left(\left|{x}\right| \right)} - \frac{3 \log{\left(\left|{x - \frac{1}{2}}\right| \right)}}{4}\) + C.
  1. Work on each open interval where the original integrand is real and defined.
  2. Factor the denominator, perform polynomial division if needed, and solve for the partial-fraction coefficients.
  3. The integrand rewrites as \(\frac{1}{2} - \frac{3}{2 \left(2 x - 1\right)} + \frac{1}{x}\). Apply ∫xᵖdx=x^(p+1)/(p+1) for p≠−1, ∫dx/x=log|x|, and the elementary exponential/trigonometric rules.
  4. An antiderivative is F(x)=\(\frac{x}{2} + \log{\left(\left|{x}\right| \right)} - \frac{3 \log{\left(\left|{x - \frac{1}{2}}\right| \right)}}{4}\).
  5. Differentiate this expression to check that F′(x)=\(\frac{1 - x^{2}}{x \left(1 - 2 x\right)}\). Add an arbitrary constant C on each domain component.

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