Evaluate the Integral

Problem

(∫_0^1)((x−6)/(x2−6*x+8)*d(x))

Solution

1. Factor the denominator of the integrand.

x2−6*x+8=(x−4)*(x−2)

2. Set up the partial fraction decomposition for the integrand.

(x−6)/((x−4)*(x−2))=A/(x−4)+B/(x−2)

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

x−6=A*(x−2)+B*(x−4)

4. Substitute x=4 to find A

4−6=A*(4−2)

−2=2*A

A=−1

5. Substitute x=2 to find B

2−6=B*(2−4)

−4=−2*B

B=2

6. Rewrite the integral using the partial fractions.

(∫_0^1)(((−1)/(x−4)+2/(x−2))*d(x))

7. Integrate each term using the natural logarithm rule.

[−ln(x−4)+2*ln(x−2)]10

8. Evaluate the definite integral at the upper and lower limits.

(−ln(1−4)+2*ln(1−2))−(−ln(0−4)+2*ln(0−2))

9. Simplify the logarithmic expressions.

(−ln(3)+2*ln(1))−(−ln(4)+2*ln(2))

−ln(3)+0+ln(4)−2*ln(2)

10. Apply logarithm properties to further simplify the result.

ln(4)=2*ln(2)

−ln(3)+2*ln(2)−2*ln(2)=−ln(3)

Final Answer

(∫_0^1)((x−6)/(x2−6*x+8)*d(x))=−ln(3)

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