Evaluate the Integral

Problem

(∫_^)(1/(x2√(,4−x2))*d(x))

Solution

1. Identify the trigonometric substitution required for the radical form √(,a2−x2) where a=2

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

3. Simplify the radical expression using the identity 1−sin2(θ)=cos2(θ)

√(,4−(2*sin(θ))2)=√(,4−4*sin2(θ))

√(,4*(1−sin2(θ)))=2*cos(θ)

4. Rewrite the integral in terms of θ by substituting x d(x) and the simplified radical.

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

5. Simplify the integrand by canceling common terms.

(∫_^)(1/(4*sin2(θ))*d(θ))

6. Apply the trigonometric identity 1/sin2(θ)=csc2(θ)

1/4*(∫_^)(csc2(θ)*d(θ))

7. Integrate using the standard integral (∫_^)(csc2(θ)*d(θ))=−cot(θ)+C

−1/4*cot(θ)+C

8. Convert back to the variable x using the relationship sin(θ)=x/2 which implies a right triangle with opposite side x hypotenuse 2 and adjacent side √(,4−x2)

cot(θ)=adjacent/opposite=√(,4−x2)/x

9. Substitute the expression for cot(θ) into the result.

−1/4⋅√(,4−x2)/x+C

Final Answer

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

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