91Ó°ÊÓ

Integration by substitution is a powerful technique used to simplify integrals. It involves changing variables to make the integral easier to evaluate.
When faced with a complicated integrand, identify a part of it that can be substituted with a new variable. This is usually a function and its derivative. Here, let's see how this technique is applied:
For the integral \(\begin{aligned} \int_{1}^{3} 2x e^{x^{2}} dx \end{aligned}\), note that \(2x dx\) is the derivative of \(x^2\). This hints at using substitution. Let \(u = x^2\), which gives \(du = 2x dx\).
The integral now becomes \(\begin{aligned} \int e^u du \end{aligned}\), which is straightforward. After integrating, don't forget to change back to the original variable by substituting \(u = x^2\).

A definite integral has both upper and lower limits. This means after finding the antiderivative, you evaluate it at the given limits and subtract.
Let's break it down using our transformed integral: \(\begin{aligned} \bigg[ e^u \bigg]_{x=1}^{x=3} \end{aligned}\)
After applying substitution and integrating, we get \(\begin{aligned} e^{u} \bigg]_{1}^{9} \end{aligned}\).
To evaluate it, substitute the limits \(1\) and \(9\) into \(u\), giving \(\begin{aligned} e^9 - e^1 \end{aligned}\).
This final subtraction step yields the value of the definite integral.

An exponential function is a mathematical function of the form \(e^x\), where \(e\) is the base of the natural logarithm, approximately equal to \(2.718\).
Exponential functions grow rapidly and have unique properties that make them essential in calculus.
In the given problem, \(e^{x^2}\) dominates the integrand. Knowing how to integrate exponential functions, especially with substitution, simplifies solving these types of integrals.
The exponent in \(e^{x^2}\) can be managed by changing variables, as seen in our substitution step. These techniques allow handling integrals of exponential functions more efficiently.