NCERT Class 12 · Ch 7 Exercise 7.4 · Q2
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.4 Question 2
Evaluate \(\int \frac{1}{\sqrt{4 x^{2} + 1}}\, dx\).
💡 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: \(\frac{\operatorname{asinh}{\left(2 x \right)}}{2}\) + C.
- Work on each open interval where the integrand is real and defined.
- u=2x; use ∫du/√(1+u²)=asinh u=log(u+√(1+u²)).
- An antiderivative is F(x)=\(\frac{\operatorname{asinh}{\left(2 x \right)}}{2}\).
- Differentiate this expression to check that F′(x)=\(\frac{1}{\sqrt{4 x^{2} + 1}}\). Add an arbitrary constant C on each domain component.
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