Find the Integral square root of 9-x^2

Problem

(∫_^)(√(,9−x^2)*d(x))

Solution

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

2. Substitute x=3*sin(θ) which implies d(x)=3*cos(θ)*d(θ)

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

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

4. Rewrite the integral in terms of θ

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

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

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

6. Expand using the double-angle identity sin(2*θ)=2*sin(θ)*cos(θ)

9/2*θ+9/2*sin(θ)*cos(θ)+C

7. Back-substitute using θ=arcsin(x/3) sin(θ)=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

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

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