Evaluate the Integral

Problem

(∫_^)(√(,x^2+4)*d(x))

Solution

1. Identify the integral form as √(,x^2+a^2) where a=2 suggesting the trigonometric substitution x=2*tan(θ)

2. Differentiate the substitution to find d(x)=2*sec^2(θ)*d(θ)

3. Substitute the expressions into the integral to get (∫_^)(√(,4*tan^2(θ)+4)⋅2*sec^2(θ)*d(θ))

4. Simplify the integrand using the identity 1+tan^2(θ)=sec^2(θ) resulting in (∫_^)(2*sec(θ)⋅2*sec^2(θ)*d(θ))=4*(∫_^)(sec^3(θ)*d(θ))

5. Apply the reduction formula for the integral of sec^3(θ) which is (∫_^)(sec^3(θ)*d(θ))=1/2*(sec(θ)*tan(θ)+ln(sec(θ)+tan(θ)))

6. Multiply by the constant factor to get 2*sec(θ)*tan(θ)+2*ln(sec(θ)+tan(θ))+C

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

8. Simplify the final expression, noting that the constant terms from the logarithm can be absorbed into the constant of integration C

Final Answer

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

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