NCERT Class 12 · Ch 7 Exercise 7.9 · Q4
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.9 Question 4
Evaluate \(\int\limits_{0}^{2} x \sqrt{x + 2}\, dx\).
💡 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: \(\frac{16 \sqrt{2}}{15} + \frac{32}{15}\)
- Work on each open interval where the original integrand is real and defined.
- Put x+2=t². Then x=t²−2, dx=2t dt and limits √2,2.
- An antiderivative is F(x)=\(\frac{2 \left(x + 2\right)^{\frac{5}{2}}}{5} - \frac{4 \left(x + 2\right)^{\frac{3}{2}}}{3}\).
- At the upper and lower limits, F(b)=\(\frac{32}{15}\) and F(a)=\(- \frac{16 \sqrt{2}}{15}\).
- By the fundamental theorem (using one-sided limits for any integrable endpoint), I=F(b)−F(a)=\(\frac{16 \sqrt{2}}{15} + \frac{32}{15}\).
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