Definite Integration
Grade None
Question:
<p>The value of the definite integral <span class="math-tex">\(\int_\limits{-1}^{1} e^{-x^{4}}\left(1+\ln (x+\sqrt{x^{2}+1})+5 x^{3}-4 x^{4}\right) d x\)</span> is equal to</p>
<p style="display:inline">4e</p>
<p style="display:inline"><span class="math-tex">\(\frac{4}{e}\)</span></p>
<p style="display:inline">2e</p>
<p style="display:inline"><span class="math-tex">\(\frac{2}{e}\)</span></p>
Step-by-Step Solution
Key Concept: Exploit the property of odd functions over a symmetric interval to eliminate complex terms and recognize the remaining integrand as a perfect derivative of the form d/dx(x * f(x)).
<p>As, f(x) <span class="math-tex">\(=1+\ln (x+\sqrt{x^{2}+1})+5 x^{3}-4 x^{4}\)</span><br />
Cleary, <span class="math-tex">\(\ln (\sqrt{x^{2}+1}+x), 5 x^{3}\)</span> are odd functions, so<br />
<span class="math-tex">\(I=\int_{-1}^{1}\left(1-4 x^{4}\right) e^{-x^{4}} d x=2 \int_{0}^{1}\left(1-4 x^{4}\right) e^{-x^{4}} d x\)</span><br />
<span class="math-tex">\(=2 \int_{0}^{1} \frac{d}{d x}\left(x \cdot e^{-x^{4}}\right) d x\)</span><br />
<span class="math-tex">\(=2\left(\frac{x}{e^{x^{4}}}\right)_{0}^{1}=2\left(\frac{1}{e}-0\right)=\frac{2}{e}\)</span></p>
Correct Answer: D