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
This expression leads to an anti-derivative involving a logarithm, as the quadratic in the denominator does not factor nicely.
Step 2
B. \( \frac{x + 2}{x^2 - 4x - 5} \)
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Answer
Similar to A, this expression also results in an anti-derivative that contains a logarithmic term after partial fraction decomposition.
Step 3
C. \( \frac{x - 1}{x^3 - x^2 + x - 1} \)
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Answer
This expression can be simplified and its anti-derivative computed without involving logarithms, as it can be factored.
Step 4
D. \( \frac{x + 1}{x^3 - x^2 + x - 1} \)
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Answer
Like expression C, this one can also lead to an anti-derivative that does not include logarithmic terms.
