Suppose you want to evaluate the integral ∫
2x−4

(x−3)(x+1)
dx using the method of partial fractions. The partial fraction decomposition has the form:

2x−4

(x−3)(x+1)
=
A

x−3
+
B

x+1
for some constants A and B.

Find the values of the constants A and B using either the "Method of Equating Coefficients" or the "Method of Strategic Substitution" (see Example 3.29 in the textbook
Links to an external site.
).

What is the value of the constant B ?

(Answer with a numerical value in decimal form, e.g. 0.5 rather than 1/2.)

Question

Suppose you want to evaluate the integral ∫
2x−4

(x−3)(x+1)
dx using the method of partial fractions. The partial fraction decomposition has the form:

2x−4

(x−3)(x+1)
=
A

x−3
+
B

x+1
  for some constants A  and B.

Find the values of the constants A  and B using either the "Method of Equating Coefficients" or the "Method of Strategic Substitution" (see Example 3.29 in the textbook
Links to an external site.
).

What is the value of the constant B ?

(Answer with a numerical value in decimal form, e.g. 0.5 rather than 1/2.)

BlackTom AI · Accepted Answer

We are decomposing  (2x - 4)/[(x - 3)(x + 1)] into A/(x - 3) + B/(x + 1).

Step 1: Write the identity by clearing denominators:
2x - 4 = A(x + 1) + B(x - 3).

Step 2: Expand the right-hand side: A(x + 1) + B(x - 3) = Ax + A + Bx - 3B = (A + B)x + (A - 3B).

Step 3: Match coefficients with the left-hand side, which is 2x - 4. This gives a system of linear equations:
- For the x-coefficient: A + B = 2.
- For the constant term: A - 3B = -4.

Step 4: Solve the system. From A = 2 - B, substitute into the second equation:
(2 - B) - 3B = -4 ⇒ 2 - 4B = -4 ⇒ -4B = -6 ⇒ B = (-6)/(-4) = 1.5.

Step 5: (Optional consistency check) With B = 1.5, A = 2 - B = 0.5. Check: A + B = 0.5 + 1.5 = 2 (ok), A - 3B = 0.5 - 4.5 = -4 (ok).

Step 6: Conclusion from the derived values: B equals 1.5.

MAT136H5 S 2025 - All Sections 3.4 Preparation Check

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