Definite Integration
Properties of Definite Integrals
Grade 12
Question:
<p>If \(\int_{-a}^{a} \left( e^x + \cos x \ln\left( x + \sqrt{1+x^2} \right) \right) dx > 2\), then possible value of \(a\) can be:</p>
<p>(a) 1</p>
<p>(b) 2</p>
<p>(c) 3</p>
<p>(d) 4</p>
Step-by-Step Solution
Key Concept: Recognize odd and even functions to simplify definite integrals and solve inequalities.
<p>The first integrand $e^x$ is even and $\cos x \ln(x + \sqrt{1+x^2})$ is odd. So $\int_{-a}^{a} e^x dx = 2\int_0^a e^x dx = 2(e^a - 1) > 2$, which gives $e^a > 2$, so $a > \ln 2 \approx 0.69$. Values 3 and 4 satisfy this.</p>
Correct Answer: c, d