Find the Integral square root of 1-x^2

Problem

(∫_^)(√(,1−x2)*d(x))

Solution

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

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

3. Simplify the radical using the identity 1−sin2(θ)=cos2(θ) so √(,1−sin2(θ))=cos(θ)

4. Rewrite the integral in terms of θ

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

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

(∫_^)((1+cos(2*θ))/2*d(θ))=1/2*θ+1/4*sin(2*θ)+C

6. Use the double-angle identity sin(2*θ)=2*sin(θ)*cos(θ) to prepare for back-substitution:

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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