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NCERT Class 12 · Ch 7 Exercise 7.5 · Q11

NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.5 Question 11

Evaluate \(\int \frac{5 x}{\left(x + 1\right) \left(x^{2} - 4\right)}\, dx\).

💡 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: \(\frac{5 \log{\left(\left|{x - 2}\right| \right)}}{6} + \frac{5 \log{\left(\left|{x + 1}\right| \right)}}{3} - \frac{5 \log{\left(\left|{x + 2}\right| \right)}}{2}\) + 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 \(- \frac{5}{2 \left(x + 2\right)} + \frac{5}{3 \left(x + 1\right)} + \frac{5}{6 \left(x - 2\right)}\). 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)=\(\frac{5 \log{\left(\left|{x - 2}\right| \right)}}{6} + \frac{5 \log{\left(\left|{x + 1}\right| \right)}}{3} - \frac{5 \log{\left(\left|{x + 2}\right| \right)}}{2}\).
  5. Differentiate this expression to check that F′(x)=\(\frac{5 x}{\left(x + 1\right) \left(x^{2} - 4\right)}\). Add an arbitrary constant C on each domain component.

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