Question

The value of \( \displaystyle \int_{-1}^{1} x^2 e^{x^3} \, dx \), where \( [t] \) denotes the greatest integer \( \leq t \), is :

• A. (e + 1)/3
• B. 1/(3e)
• C. (e - 1)/(3e)
• D. (e + 1)/(3e)

Hint:

When you see a greatest integer function $[t]$ inside an integral, always split the limits at points where the expression inside the brackets becomes an integer. For $\int_{-1}^{1} [x^2] e^{x^3} dx$: \begin{itemize} \item On the interval $x \in (-1, 1)$, we have $0 \leq x^2<1$, which means $[x^2] = 0$. \item Since the integrand is zero almost everywhere on the interval, the integral value is simply {0}. \end{itemize}

Answer 1

The Correct Option is C

To solve the integral \(\int_{-1}^{1} x^2 e^{x^3} \, dx\), we can make use of properties of definite integrals and symmetry. Let's go through the solution step-by-step:

1. First, notice that the function \(x^2 e^{x^3}\) is an even function. An even function satisfies \(f(-x) = f(x)\). Here, \((-x)^2 e^{(-x)^3} = x^2 e^{-x^3}\), which is not symmetric, but \((-x)^2 = x^2\), showing the \(x^2\) part is even.
2. The function \((e^{x^3} = e^{-x^3})\) reduces the symmetry condition to: \(x^2\) is even.
3. The integral of an odd function over a symmetric interval around zero is zero. However, the combination of \(x^2\) and \(e^{x^3}\) makes the function as a whole not completely odd due to its integration properties.
4. To solve the integral, we use substitution. Let \(u = x^3\), then \(\frac{du}{dx} = 3x^2\) or dx = \frac{du}{3x^2}.
5. Substitute: \(\int_{-1}^{1} x^2 e^{x^3} \, dx = \int_{-1}^{1} x^2 e^u \frac{du}{3x^2}\), which simplifies to \(\frac{1}{3} \int_{-1}^{1} e^u \, du\).
6. Calculate this integral by changing the limits: When \(x = -1\), \(u = (-1)^3 = -1\), and when \(x = 1\), \(u = 1^3 = 1\).
7. Use: \(\frac{1}{3} [e^u]_{-1}^{1} = \frac{1}{3} (e^1 - e^{-1})\) which is \(\frac{1}{3} (e - \frac{1}{e})\).
8. simplify: Simplify \(\frac{1}{3} (e - \frac{1}{e})\) as \(\frac{e^2 - 1}{3e}\).

Thus, the value of the integral \(\int_{-1}^{1} x^2 e^{x^3} \, dx\) is \(\frac{e - 1}{3e}\). Therefore, the correct answer is:

(e - 1)/(3e)