Which of the following is an anti-derivative of
\[
\frac{1}{4x^{2}+1}
\]?
A - HSC - SSCE Mathematics Extension 1 - Question 3 - 2020 - Paper 1
Question 3
Which of the following is an anti-derivative of
\[
\frac{1}{4x^{2}+1}
\]?
A. 2 tan$^{-1} \left( \frac{x}{2} \right) + c$
B. \(\frac{1}{2} tan^{-1} \left( \frac{x}{2... show full transcript
Worked Solution & Example Answer:Which of the following is an anti-derivative of
\[
\frac{1}{4x^{2}+1}
\]?
A - HSC - SSCE Mathematics Extension 1 - Question 3 - 2020 - Paper 1
Step 1
Identify the function and its form
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Answer
The given function is (\frac{1}{4x^{2} + 1}). This resembles the standard integral form (\int \frac{1}{a^{2} + x^{2}} , dx = \frac{1}{a} \tan^{-1} \left( \frac{x}{a} \right) + c) where (a = 2).
Step 2
Apply the integral formula
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Answer
Letting (a = 2), we integrate:
[
\int \frac{1}{4x^{2} + 1} , dx = \frac{1}{2} \tan^{-1} \left( \frac{x}{2} \right) + c
]
Step 3
Select the correct option
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Answer
From our integration, the correct option corresponds to B: (\frac{1}{2} tan^{-1} \left( \frac{x}{2} \right) + c).
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