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Integrate (x - 1)/((x - 3)(x - 2)) dx = (a) log(x - 3) - log(x - 2) + c (b) log(x - 3) ^ 2 - log(x - 2) + c (c) log(x - 3) + log(x - 2) + c (d) log(x - 3) ^ 2 + log(x - 2) + c jon.c?
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Integrate (x - 1)/((x - 3)(x - 2)) dx = (a) log(x - 3) - log(x - 2) + ...
Integration of (x - 1)/((x - 3)(x - 2)) dx
Integration of rational functions like (x - 1)/((x - 3)(x - 2)) can be done using partial fraction decomposition.
Partial Fraction Decomposition
To integrate the given function, we need to express it as a sum of simpler fractions. The general form of the partial fraction decomposition for the given function is:
(x - 1)/((x - 3)(x - 2)) = A/(x - 3) + B/(x - 2)
Solving for A and B
Multiplying both sides by ((x - 3)(x - 2)), we get:
x - 1 = A(x - 2) + B(x - 3)
By substituting x = 3, we get A = 2. By substituting x = 2, we get B = -1.
Therefore, the partial fraction decomposition is:
(x - 1)/((x - 3)(x - 2)) = 2/(x - 3) - 1/(x - 2)
Integrating the Decomposed Fractions
Integrating the decomposed fractions, we get:
∫(2/(x - 3) - 1/(x - 2)) dx = 2∫(1/(x - 3)) dx - ∫(1/(x - 2)) dx
This simplifies to:
= 2log| x - 3 | - log| x - 2 | + C
Therefore, the correct answer is option (a) log(x - 3) - log(x - 2) + C.
Integration of rational functions like (x - 1)/((x - 3)(x - 2)) can be done using partial fraction decomposition.
Partial Fraction Decomposition
To integrate the given function, we need to express it as a sum of simpler fractions. The general form of the partial fraction decomposition for the given function is:
(x - 1)/((x - 3)(x - 2)) = A/(x - 3) + B/(x - 2)
Solving for A and B
Multiplying both sides by ((x - 3)(x - 2)), we get:
x - 1 = A(x - 2) + B(x - 3)
By substituting x = 3, we get A = 2. By substituting x = 2, we get B = -1.
Therefore, the partial fraction decomposition is:
(x - 1)/((x - 3)(x - 2)) = 2/(x - 3) - 1/(x - 2)
Integrating the Decomposed Fractions
Integrating the decomposed fractions, we get:
∫(2/(x - 3) - 1/(x - 2)) dx = 2∫(1/(x - 3)) dx - ∫(1/(x - 2)) dx
This simplifies to:
= 2log| x - 3 | - log| x - 2 | + C
Therefore, the correct answer is option (a) log(x - 3) - log(x - 2) + C.
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Integrate (x - 1)/((x - 3)(x - 2)) dx = (a) log(x - 3) - log(x - 2) + c (b) log(x - 3) ^ 2 - log(x - 2) + c (c) log(x - 3) + log(x - 2) + c (d) log(x - 3) ^ 2 + log(x - 2) + c jon.c?
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