NCERT Class 12 · Ch 7 Exercise 7.1 · Q15
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.1 Question 15
Evaluate \(\int \sqrt{x} \left(3 x^{2} + 2 x + 3\right)\, dx\).
💡 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: \(\frac{6 x^{\frac{7}{2}}}{7} + \frac{4 x^{\frac{5}{2}}}{5} + 2 x^{\frac{3}{2}}\) + C.
- Work on each open interval where the original integrand is real and defined.
- Expand and use linearity, then integrate each power or elementary term.
- The integrand rewrites as \(3 x^{\frac{5}{2}} + 2 x^{\frac{3}{2}} + 3 \sqrt{x}\). 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)=\(\frac{6 x^{\frac{7}{2}}}{7} + \frac{4 x^{\frac{5}{2}}}{5} + 2 x^{\frac{3}{2}}\).
- Differentiate this expression to check that F′(x)=\(\sqrt{x} \left(3 x^{2} + 2 x + 3\right)\). Add an arbitrary constant C on each domain component.
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