It is given that $f(x)$ is a non-zero even function and $g(x)$ is a non-zero odd function - HSC - SSCE Mathematics Extension 2 - Question 7 - 2017 - Paper 1
Question 7
It is given that $f(x)$ is a non-zero even function and $g(x)$ is a non-zero odd function.
Which expression is equal to $$\int_{-a}^{a} f(x) + g(x) \, dx$$?
Worked Solution & Example Answer:It is given that $f(x)$ is a non-zero even function and $g(x)$ is a non-zero odd function - HSC - SSCE Mathematics Extension 2 - Question 7 - 2017 - Paper 1
Step 1
Evaluate the integral of the even function: $\int_{-a}^{a} f(x) \, dx$
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Answer
Since f(x) is an even function, we can use the property of even functions that states:
∫−aaf(x)dx=2∫0af(x)dx.
Step 2
Evaluate the integral of the odd function: $\int_{-a}^{a} g(x) \, dx$
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Answer
For the odd function g(x), the property states:
∫−aag(x)dx=0.
Step 3
Combine the results
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Answer
Hence, we can combine the two results:
∫−aa(f(x)+g(x))dx=∫−aaf(x)dx+∫−aag(x)dx=2∫0af(x)dx+0.
Thus, the expression ∫−aaf(x)+g(x)dx is equal to the expression:
2∫0af(x)dx.
Step 4
Identify the correct option
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Answer
Based on the evaluation, the answer corresponds to option A:
2∫0af(x)dx.
