Of the following expressions, which one need NOT contain a term involving a logarithm in its anti-derivative?
A - HSC - SSCE Mathematics Extension 2 - Question 4 - 2022 - Paper 1
Question 4
Of the following expressions, which one need NOT contain a term involving a logarithm in its anti-derivative?
A. \( \frac{x + 2}{x^2 + 4x + 5} \)
B. \( \frac{x + 2... show full transcript
Worked Solution & Example Answer:Of the following expressions, which one need NOT contain a term involving a logarithm in its anti-derivative?
A - HSC - SSCE Mathematics Extension 2 - Question 4 - 2022 - Paper 1
Step 1
A. \( \frac{x + 2}{x^2 + 4x + 5} \)
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Answer
To find the anti-derivative of this expression, we observe that the denominator can be factored into irreducible quadratics, which may introduce terms involving logarithms.
Step 2
B. \( \frac{x + 2}{x^2 - 4x - 5} \)
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Answer
This denominator can be factored as ( (x - 5)(x + 1) ), which will require a logarithm in the anti-derivative because it introduces poles at the roots of the denominator.
Step 3
C. \( x - 1 \)
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Answer
The function ( x - 1 ) is a polynomial. Its anti-derivative is simply ( \frac{x^2}{2} - x + C ), which does NOT involve any logarithmic terms.
Step 4
D. \( \frac{x + 1}{x^3 - x^2 + x - 1} \)
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Answer
This expression has a polynomial denominator, which may not necessarily simplify to prevent logarithmic terms, depending on its roots.
