NCERT Class 12 · Ch 7 Exercise 7.4 · Q19
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.4 Question 19
Evaluate \(\int \frac{6 x + 7}{\sqrt{\left(x - 5\right) \left(x - 4\right)}}\, dx\).
💡 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: \(6 \sqrt{\left(x - 5\right) \left(x - 4\right)} + 34 \log{\left(\left|{x + \sqrt{\left(x - 5\right) \left(x - 4\right)} - \frac{9}{2}}\right| \right)}\) + C.
- Work on each open interval where the integrand is real and defined.
- Write 6x+7=3(2x−9)+34; the first part is a radicand derivative and the second a standard logarithm.
- An antiderivative is F(x)=\(6 \sqrt{\left(x - 5\right) \left(x - 4\right)} + 34 \log{\left(\left|{x + \sqrt{\left(x - 5\right) \left(x - 4\right)} - \frac{9}{2}}\right| \right)}\).
- Differentiate this expression to check that F′(x)=\(\frac{6 x + 7}{\sqrt{\left(x - 5\right) \left(x - 4\right)}}\). Add an arbitrary constant C on each domain component.
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