NCERT Class 12 · Ch 7 Exercise 7.4 · Q23
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.4 Question 23
Evaluate \(\int \frac{5 x + 3}{\sqrt{x^{2} + 4 x + 10}}\, dx\).
💡 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: \(5 \sqrt{x^{2} + 4 x + 10} - 7 \operatorname{asinh}{\left(\frac{\sqrt{6} \left(x + 2\right)}{6} \right)}\) + C.
- Work on each open interval where the integrand is real and defined.
- Write 5x+3=5(x+2)−7; complete the square (x+2)²+6.
- An antiderivative is F(x)=\(5 \sqrt{x^{2} + 4 x + 10} - 7 \operatorname{asinh}{\left(\frac{\sqrt{6} \left(x + 2\right)}{6} \right)}\).
- Differentiate this expression to check that F′(x)=\(\frac{5 x + 3}{\sqrt{x^{2} + 4 x + 10}}\). Add an arbitrary constant C on each domain component.
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