NCERT Class 12 · Ch 7 Exercise 7.7 · Q11
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.7 Question 11
Evaluate \(\int \sqrt{x^{2} - 8 x + 7}\, dx\).
💡 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: D: \(\frac{\left(x - 4\right) \sqrt{\left(x - 4\right)^{2} - 9}}{2} - \frac{9 \log{\left(\left|{x + \sqrt{\left(x - 4\right)^{2} - 9} - 4}\right| \right)}}{2}\) + C.
- Work on each open interval where the original integrand is real and defined.
- Complete the square: x²−8x+7=(x−4)²−9.
- Put u=x−4; the root formula gives [u√(u²−9)−9log|u+√(u²−9)|]/2.
- An antiderivative is F(x)=\(\frac{\left(x - 4\right) \sqrt{\left(x - 4\right)^{2} - 9}}{2} - \frac{9 \log{\left(\left|{x + \sqrt{\left(x - 4\right)^{2} - 9} - 4}\right| \right)}}{2}\).
- Differentiate this expression to check that F′(x)=\(\sqrt{x^{2} - 8 x + 7}\). Add an arbitrary constant C on each domain component.
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