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NCERT Class 12 · Ch 7 Exercise 7.5 · Q9

NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.5 Question 9

Evaluate \(\int \frac{3 x + 5}{x^{3} - x^{2} - x + 1}\, dx\).

💡 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: \(- \frac{\log{\left(\left|{x - 1}\right| \right)}}{2} + \frac{\log{\left(\left|{x + 1}\right| \right)}}{2} - \frac{4}{x - 1}\) + 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 \(\frac{1}{2 \left(x + 1\right)} - \frac{1}{2 \left(x - 1\right)} + \frac{4}{\left(x - 1\right)^{2}}\). 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)=\(- \frac{\log{\left(\left|{x - 1}\right| \right)}}{2} + \frac{\log{\left(\left|{x + 1}\right| \right)}}{2} - \frac{4}{x - 1}\).
  5. Differentiate this expression to check that F′(x)=\(\frac{3 x + 5}{x^{3} - x^{2} - x + 1}\). Add an arbitrary constant C on each domain component.

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