Evaluate the Integral

Problem

(∫_4^5)((x^3−3*x^2−9)/(x^3−3*x^2)*d(x))

Solution

1. Perform polynomial division to simplify the integrand by splitting the numerator.

(x^3−3*x^2−9)/(x^3−3*x^2)=(x^3−3*x^2)/(x^3−3*x^2)−9/(x^3−3*x^2)

(x^3−3*x^2−9)/(x^3−3*x^2)=1−9/(x^2*(x−3))

2. Decompose into partial fractions for the remaining rational term.

9/(x^2*(x−3))=A/x+B/(x^2)+C/(x−3)

9=A*x*(x−3)+B*(x−3)+C*x^2

3. Solve for the constants by substituting values for x

Let *x=0⇒9=−3*B⇒B=−3

Let *x=3⇒9=9*C⇒C=1

Equate *x^2* coefficients⇒0=A+C⇒A=−1

4. Rewrite the integral using the simplified expression and partial fractions.

(∫_4^5)((1−((−1)/x+(−3)/(x^2)+1/(x−3)))*d(x))

(∫_4^5)((1+1/x+3/(x^2)−1/(x−3))*d(x))

5. Find the antiderivative of each term.

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

6. Apply the Fundamental Theorem of Calculus by evaluating at the boundaries x=5 and x=4

(5+ln(5)−3/5−ln(2))−(4+ln(4)−3/4−ln(1))

(4.4+ln(5/2))−(3.25+ln(4))

1.15+ln(5/8)

Final Answer

(∫_4^5)((x^3−3*x^2−9)/(x^3−3*x^2)*d(x))=23/20+ln(5/8)

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