Evaluate the Integral

Problem

(∫_^)((5−x)/(2*x2+x−1)*d(x))

Solution

1. Factor the denominator of the integrand.

2*x2+x−1=(2*x−1)*(x+1)

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

(5−x)/((2*x−1)*(x+1))=A/(2*x−1)+B/(x+1)

3. Solve for the constants A and B by clearing the denominators.

5−x=A*(x+1)+B*(2*x−1)

4. Substitute x=−1 to find B

5−(−1)=B*(2*(−1)−1)

6=−3*B

B=−2

5. Substitute x=1/2 to find A

5−1/2=A*(1/2+1)

9/2=3/2*A

A=3

6. Rewrite the integral using the partial fractions.

(∫_^)((5−x)/(2*x2+x−1)*d(x))=(∫_^)(3/(2*x−1)*d(x))+(∫_^)((−2)/(x+1)*d(x))

7. Integrate each term separately using the substitution rule for linear denominators.

(∫_^)(3/(2*x−1)*d(x))=3/2*ln(2*x−1)

(∫_^)((−2)/(x+1)*d(x))=−2*ln(x+1)

8. Combine the results and add the constant of integration C

3/2*ln(2*x−1)−2*ln(x+1)+C

Final Answer

(∫_^)((5−x)/(2*x2+x−1)*d(x))=3/2*ln(2*x−1)−2*ln(x+1)+C

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