Evaluate the Integral

Problem

(∫_^)(1/√(,4−9*x2)*d(x))

Solution

1. Identify the form of the integral as a standard inverse trigonometric integral, specifically (∫_^)(1/√(,a2−u2)*d(u))=arcsin(u/a)+C

2. Rewrite the expression in the denominator to match the form a2−u2 by expressing 4 as 2 and 9*x2 as (3*x)2

(∫_^)(1/√(,2−(3*x)2)*d(x))

3. Substitute u=3*x which implies d(u)=3*d(x) or d(x)=1/3*d(u)

(∫_^)(1/√(,2−u2)⋅1/3*d(u))

4. Factor out the constant 1/3 from the integral.

1/3*(∫_^)(1/√(,2−u2)*d(u))

5. Apply the formula for the arcsine integral where a=2

1/3*arcsin(u/2)+C

6. Back-substitute u=3*x to obtain the final result in terms of x

1/3*arcsin((3*x)/2)+C

Final Answer

(∫_^)(1/√(,4−9*x2)*d(x))=1/3*arcsin((3*x)/2)+C

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