Definite Integration
Properties of Definite Integrals
Grade 12
Question:
<p>Which of the following function(s) is/are even?</p><p>(A) \(f(x) = \int_0^x \ln(t^2+1)t \, dt\)</p><p>(B) \(g(x) = \int_0^x \frac{(2t-1)t}{t} \, dt\)</p><p>(C) \(h(x) = \int_0^x \frac{1+t}{1+t^2} \, dt\)</p><p>(D) \(l(x) = \int_0^x \ln(1+t) \, dt\)</p>
<p>(A) f(x) = \(\int_0^x \ln(t^2+1)t \, dt\)</p>
<p>(B) g(x) = \(\int_0^x \frac{(2t-1)t}{t} \, dt\)</p>
<p>(C) h(x) = \(\int_0^x \frac{1+t}{1+t^2} \, dt\)</p>
<p>(D) l(x) = \(\int_0^x \ln(1+t) \, dt\)</p>
Step-by-Step Solution
Key Concept: An even function satisfies f(-x) = f(x). When integrating an odd function from 0 to x, the result is even.
<p><strong>Analysis of even functions:</strong> A function is even if $f(-x) = f(x)$.</p><p>For (A): $f(x) = \int_0^x \ln(t^2+1)t \, dt$, since $\ln(t^2+1)t$ is an odd function, the integral evaluates to an even function.</p><p>For (B), (C), (D): The integrands are not odd functions, so these do not produce even functions.</p>
Correct Answer: A