Evaluate the Integral

Problem

(∫_^)(√(,1−4*x2)*d(x))

Solution

1. Identify the form of the integrand as √(,a2−u2) which suggests a trigonometric substitution where a=1 and u=2*x

2. Substitute 2*x=sin(θ) which implies x=1/2*sin(θ) and d(x)=1/2*cos(θ)*d(θ)

3. Simplify the square root term using the identity 1−sin2(θ)=cos2(θ)

√(,1−(2*x)2)=√(,1−sin2(θ))=cos(θ)

4. Rewrite the integral in terms of θ by substituting the expressions for the square root and d(x)

(∫_^)(cos(θ)⋅1/2*cos(θ)*d(θ))=1/2*(∫_^)(cos2(θ)*d(θ))

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

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

6. Integrate with respect to θ

1/4*(θ+1/2*sin(2*θ))+C

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

1/4*θ+1/4*sin(θ)*cos(θ)+C

8. Back-substitute to return to the variable x using θ=arcsin(2*x) sin(θ)=2*x and cos(θ)=√(,1−4*x2)

1/4*arcsin(2*x)+1/4*(2*x)√(,1−4*x2)+C

9. Simplify the final expression.

1/4*arcsin(2*x)+x/2√(,1−4*x2)+C

Final Answer

(∫_^)(√(,1−4*x2)*d(x))=(x√(,1−4*x2))/2+arcsin(2*x)/4+C

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