Evaluate the Integral

Problem

(∫_^)(1/√(,4+x^2)*d(x))

Solution

1. Identify the form of the integral as a trigonometric substitution case involving √(,a^2+x^2) where a=2

2. Substitute x=2*tan(θ) which implies d(x)=2*sec^2(θ)*d(θ)

3. Simplify the radical expression using the identity 1+tan^2(θ)=sec^2(θ)

√(,4+x^2)=√(,4+4*tan^2(θ))

√(,4+x^2)=√(,4*(1+tan^2(θ)))

√(,4+x^2)=2*sec(θ)

4. Rewrite the integral in terms of θ

(∫_^)((2*sec^2(θ))/(2*sec(θ))*d(θ))

(∫_^)(sec(θ)*d(θ))

5. Integrate the secant function.

ln(sec(θ)+tan(θ))+C

6. Back-substitute to return to the variable x using the relationships tan(θ)=x/2 and sec(θ)=√(,4+x^2)/2

ln(√(,4+x^2)/2+x/2)+C

7. Simplify the logarithmic expression by absorbing the constant denominator into the constant of integration C

ln(x+√(,4+x^2))−ln(2)+C

ln(x+√(,4+x^2))+C

Final Answer

(∫_^)(1/√(,4+x^2)*d(x))=ln(x+√(,4+x^2))+C

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