Evaluate the Integral

Problem

(∫_^)((x^2)/√(,9−x^2)*d(x))

Solution

1. Identify the form of the integrand, which contains √(,a^2−x^2) where a=3

2. Apply the substitution x=3*sin(θ) which implies d(x)=3*cos(θ)*d(θ)

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

√(,9−(3*sin(θ))^2)=√(,9*(1−sin^2(θ)))=3*cos(θ)

4. Substitute all terms into the integral:

(∫_^)(((3*sin(θ))^2)/(3*cos(θ))⋅3*cos(θ)*d(θ))=(∫_^)(9*sin^2(θ)*d(θ))

5. Use the power-reduction identity sin^2(θ)=(1−cos(2*θ))/2

9/2*(∫_^)((1−cos(2*θ))*d(θ))=9/2*(θ−sin(2*θ)/2)+C

6. Apply the double-angle identity sin(2*θ)=2*sin(θ)*cos(θ) to simplify the result:

9/2*θ−9/2*sin(θ)*cos(θ)+C

7. Back-substitute to return to the variable x using sin(θ)=x/3 θ=arcsin(x/3) and cos(θ)=√(,9−x^2)/3

9/2*arcsin(x/3)−9/2*(x/3)*(√(,9−x^2)/3)+C

Final Answer

(∫_^)((x^2)/√(,9−x^2)*d(x))=9/2*arcsin(x/3)−(x√(,9−x^2))/2+C

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