Evaluate the Integral

Problem

(∫_^)(1/√(,1−4*x2)*d(x))

Solution

1. Identify the form of the integral as a standard inverse trigonometric integral, specifically (∫_^)(1/√(,1−u2)*d(u))=arcsin(u)+C

2. Rewrite the expression inside the square root to identify the term u2

(∫_^)(1/√(,1−(2*x)2)*d(x))

3. Apply a substitution where u=2*x

4. Differentiate the substitution to find d(u)

d(u)/d(x)=2

d(u)=2*d(x)

d(x)=1/2*d(u)

5. Substitute the variables into the integral.

(∫_^)(1/√(,1−u2)⋅1/2*d(u))

6. Factor out the constant from the integral.

1/2*(∫_^)(1/√(,1−u2)*d(u))

7. Integrate using the arcsine rule.

1/2*arcsin(u)+C

8. Back-substitute u=2*x to get the final result in terms of x

1/2*arcsin(2*x)+C

Final Answer

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

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