Evaluate the Integral

Problem

(∫_0^1)((x−8)/(x2−7*x+10)*d(x))

Solution

1. Factor the denominator of the integrand.

x2−7*x+10=(x−2)*(x−5)

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

(x−8)/((x−2)*(x−5))=A/(x−2)+B/(x−5)

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

x−8=A*(x−5)+B*(x−2)

4. Substitute x=2 to find A

2−8=A*(2−5)

−6=−3*A

A=2

5. Substitute x=5 to find B

5−8=B*(5−2)

−3=3*B

B=−1

6. Rewrite the integral using the partial fractions.

(∫_0^1)((2/(x−2)−1/(x−5))*d(x))

7. Integrate each term using the natural logarithm rule.

[2*ln(x−2)−ln(x−5)]10

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

(2*ln(1−2)−ln(1−5))−(2*ln(0−2)−ln(0−5))

(2*ln(1)−ln(4))−(2*ln(2)−ln(5))

9. Simplify the expression using logarithmic properties, noting that ln(1)=0 and ln(4)=2*ln(2)

0−2*ln(2)−2*ln(2)+ln(5)

ln(5)−4*ln(2)

ln(5)−ln(16)

ln(5/16)

Final Answer

(∫_0^1)((x−8)/(x2−7*x+10)*d(x))=ln(5/16)

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