NCERT Class 12 · Ch 7 Exercise 7.2 · Q9
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.2 Question 9
Evaluate \(\int \left(4 x + 2\right) \sqrt{x^{2} + x + 1}\, dx\).
💡 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: \(\frac{4 \left(x^{2} + x + 1\right)^{\frac{3}{2}}}{3}\) + C.
- Work on each open interval where the original integrand is real and defined.
- u=x²+x+1, du=(2x+1)dx; the numerator is 2du.
- An antiderivative is F(x)=\(\frac{4 \left(x^{2} + x + 1\right)^{\frac{3}{2}}}{3}\).
- Differentiate this expression to check that F′(x)=\(\left(4 x + 2\right) \sqrt{x^{2} + x + 1}\). Add an arbitrary constant C on each domain component.
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