Question

guess an antiderivative for the integrand function. validate your guess by differentiation and then evaluate the given definite integral (hint: keep in mind the chain rule in guessing an antiderivative.)
$intlimits_{2}^{8} x e^{x^2} \\, dx$

$intlimits_{2}^{8} x e^{x^2} \\, dx = \square$
(type an exact answer in terms of $e$.)

Explanation:

Step1: Guess the antiderivative

Let's consider the function \( f(x) = xe^{x^2} \). We know that the derivative of \( e^{u} \) with respect to \( x \) is \( e^{u} \cdot u' \) by the chain rule. Let \( u = x^2 \), then \( u' = 2x \). Notice that our integrand has \( x \), which is half of \( 2x \). So we can guess that the antiderivative \( F(x) \) is \( \frac{1}{2}e^{x^2} \), because if we differentiate \( \frac{1}{2}e^{x^2} \) with respect to \( x \), using the chain rule: the derivative of \( e^{x^2} \) is \( e^{x^2} \cdot 2x \), and then multiply by \( \frac{1}{2} \), we get \( \frac{1}{2} \cdot e^{x^2} \cdot 2x = xe^{x^2} \), which matches our integrand. So the antiderivative \( F(x)=\frac{1}{2}e^{x^2} \).

Step2: Evaluate the definite integral

By the Fundamental Theorem of Calculus, \( \int_{a}^{b} f(x) dx = F(b) - F(a) \), where \( F(x) \) is an antiderivative of \( f(x) \). Here, \( a = 2 \), \( b = 8 \), and \( F(x)=\frac{1}{2}e^{x^2} \). So we need to compute \( F(8) - F(2) \).

First, compute \( F(8) \): \( F(8)=\frac{1}{2}e^{8^2}=\frac{1}{2}e^{64} \)

Then, compute \( F(2) \): \( F(2)=\frac{1}{2}e^{2^2}=\frac{1}{2}e^{4} \)

Now, subtract them: \( F(8)-F(2)=\frac{1}{2}e^{64}-\frac{1}{2}e^{4}=\frac{1}{2}(e^{64}-e^{4}) \)

Answer:

\(\frac{1}{2}(e^{64} - e^{4})\)