Evaluate the Integral integral of 1/(4-x^2) with respect to x

Problem

(∫_^)(1/(4−x2)*d(x))

Solution

1. Identify the form of the integrand as a rational function that can be decomposed using partial fractions.

2. Factor the denominator using the difference of squares formula.

1/(4−x2)=1/((2−x)*(2+x))

3. Set up the partial fraction decomposition.

1/((2−x)*(2+x))=A/(2−x)+B/(2+x)

4. Solve for the constants A and B by multiplying by the common denominator.

1=A*(2+x)+B*(2−x)

5. Substitute x=2 to find A

1=A(4)⇒A=1/4

6. Substitute x=−2 to find B

1=B(4)⇒B=1/4

7. Rewrite the integral using the partial fractions.

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

8. Integrate each term separately, noting the negative sign from the chain rule in the first term.

(∫_^)((1/4)/(2−x)*d(x))=−1/4*ln(2−x)

(∫_^)((1/4)/(2+x)*d(x))=1/4*ln(2+x)

9. Combine the logarithmic terms using the quotient rule for logarithms.

1/4*(ln(2+x)−ln(2−x))=1/4*ln((2+x)/(2−x))

Final Answer

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

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