NCERT Class 12 · Ch 7 Exercise 7.4 · Q21
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.4 Question 21
Evaluate \(\int \frac{x + 2}{\sqrt{x^{2} + 2 x + 3}}\, dx\).
💡 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: \(\sqrt{x^{2} + 2 x + 3} + \operatorname{asinh}{\left(\frac{\sqrt{2} \left(x + 1\right)}{2} \right)}\) + C.
- Work on each open interval where the integrand is real and defined.
- Write x+2=(x+1)+1; complete the square (x+1)²+2.
- An antiderivative is F(x)=\(\sqrt{x^{2} + 2 x + 3} + \operatorname{asinh}{\left(\frac{\sqrt{2} \left(x + 1\right)}{2} \right)}\).
- Differentiate this expression to check that F′(x)=\(\frac{x + 2}{\sqrt{x^{2} + 2 x + 3}}\). Add an arbitrary constant C on each domain component.
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