NCERT Class 12 · Ch 7 Exercise 7.5 · Q3
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.5 Question 3
Evaluate \(\int \frac{3 x - 1}{\left(x - 3\right) \left(x - 2\right) \left(x - 1\right)}\, dx\).
💡 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: \(4 \log{\left(\left|{x - 3}\right| \right)} - 5 \log{\left(\left|{x - 2}\right| \right)} + \log{\left(\left|{x - 1}\right| \right)}\) + C.
- Work on each open interval where the original integrand is real and defined.
- Factor the denominator, perform polynomial division if needed, and solve for the partial-fraction coefficients.
- The integrand rewrites as \(\frac{1}{x - 1} - \frac{5}{x - 2} + \frac{4}{x - 3}\). Apply ∫xᵖdx=x^(p+1)/(p+1) for p≠−1, ∫dx/x=log|x|, and the elementary exponential/trigonometric rules.
- An antiderivative is F(x)=\(4 \log{\left(\left|{x - 3}\right| \right)} - 5 \log{\left(\left|{x - 2}\right| \right)} + \log{\left(\left|{x - 1}\right| \right)}\).
- Differentiate this expression to check that F′(x)=\(\frac{3 x - 1}{\left(x - 3\right) \left(x - 2\right) \left(x - 1\right)}\). Add an arbitrary constant C on each domain component.
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