NCERT Class 12 · Ch 7 Exercise 7.5 · Q14
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.5 Question 14
Evaluate \(\int \frac{3 x - 1}{\left(x + 2\right)^{2}}\, dx\).
💡 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: \(3 \log{\left(\left|{x + 2}\right| \right)} + \frac{7}{x + 2}\) + 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 \(\frac{3}{x + 2} - \frac{7}{\left(x + 2\right)^{2}}\). 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)=\(3 \log{\left(\left|{x + 2}\right| \right)} + \frac{7}{x + 2}\).
- Differentiate this expression to check that F′(x)=\(\frac{3 x - 1}{\left(x + 2\right)^{2}}\). Add an arbitrary constant C on each domain component.
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