Evaluate the Integral

Problem

(∫_^)(√(,4−x^2)*d(x))

Solution

1. Identify the form of the integrand as √(,a^2−x^2) where a=2 which suggests using the trigonometric substitution x=2*sin(θ)

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

3. Substitute the expressions for x and d(x) into the integral:

(∫_^)(√(,4−(2*sin(θ))^2)⋅2*cos(θ)*d(θ))

4. Simplify the term inside the square root using the identity 1−sin^2(θ)=cos^2(θ)

(∫_^)(√(,4*(1−sin^2(θ)))⋅2*cos(θ)*d(θ))

(∫_^)(2*cos(θ)⋅2*cos(θ)*d(θ))

(∫_^)(4*cos^2(θ)*d(θ))

5. Apply the power-reduction identity cos^2(θ)=(1+cos(2*θ))/2

(∫_^)(4⋅(1+cos(2*θ))/2*d(θ))

(∫_^)((2+2*cos(2*θ))*d(θ))

6. Integrate with respect to θ

2*θ+sin(2*θ)+C

7. Use the double-angle identity sin(2*θ)=2*sin(θ)*cos(θ)

2*θ+2*sin(θ)*cos(θ)+C

8. Back-substitute to express the result in terms of x Since x=2*sin(θ) then sin(θ)=x/2 θ=arcsin(x/2) and cos(θ)=√(,1−(x/2)^2)=√(,4−x^2)/2

2*arcsin(x/2)+2*(x/2)*(√(,4−x^2)/2)+C

9. Simplify the final expression:

2*arcsin(x/2)+(x√(,4−x^2))/2+C

Final Answer

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

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