Evaluate the Integral

Problem

(∫_^)(√(,1−x2)*d(x))

Solution

1. Identify the form of the integrand as √(,a2−x2) where a=1 which suggests using the trigonometric substitution x=sin(θ)

2. Substitute x=sin(θ) which implies d(x)=cos(θ)*d(θ) and √(,1−x2)=√(,1−sin2(θ))=cos(θ)

3. Rewrite the integral in terms of θ

(∫_^)(cos(θ)*cos(θ)*d(θ))=(∫_^)(cos2(θ)*d(θ))

4. Apply the power-reduction identity cos2(θ)=(1+cos(2*θ))/2

(∫_^)((1+cos(2*θ))/2*d(θ))

5. Integrate with respect to θ

1/2*θ+1/4*sin(2*θ)+C

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

1/2*θ+1/2*sin(θ)*cos(θ)+C

7. Back-substitute using θ=arcsin(x) sin(θ)=x and cos(θ)=√(,1−x2)

1/2*arcsin(x)+1/2*x√(,1−x2)+C

Final Answer

(∫_^)(√(,1−x2)*d(x))=1/2*arcsin(x)+1/2*x√(,1−x2)+C

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