91Ó°ÊÓ

Substitution is a method that helps simplify complex integrals by changing variables. The goal is to make the integral easier to solve. In our exercise, the expression inside the square root, \(e^{2x}+1\), suggests a straightforward substitution. By letting \(u = e^{2x} + 1\), we transform the integrand.
Here’s how you can use it effectively:

• Identify a part of the integrand that complicates the integration. This might include a function inside a square root, logarithm, or exponential function.
• Choose a substitution that simplifies this expression, such as \(u = f(x)\).
• Derive \(du\) in terms of \(dx\), which is needed for changing the differential as well. In our case, \(du = 2e^{2x} dx\).

This change of variables not only simplifies the integrand but also helps in handling the differential \(dx\) within the integral.

When using substitution, converting the integration bounds is crucial to maintaining the accuracy of the integral.
Upon substituting \(u = e^{2x}+1\), the original variable, \(x\), has limits of 0 and 1. It's necessary to derive new limits in terms of \(u\).

• Start with the lower bound: When \(x=0\), substituting gives \(u=2\).
• Next, consider the upper bound: When \(x=1\), substituting gives \(u=e^2+1\).

Essentially, the substitution translates the original limits into new ones that mirror the altered variable, ensuring the definite integral evaluates correctly over the new interval \([2, e^2+1]\). Always remember to adjust the limits when you perform the substitution.

The calculation of the antiderivative is pivotal in obtaining the solution to a definite integral. Once the integral's variable is substituted and the new bounds set, the next step is to find its antiderivative.
In the reformulated integral, \(\int \frac{\sqrt{u}}{u-1} du\), simplifying and finding the antiderivative involves expressing the integrand in a more workable form. Here, it transforms into: \(u^{-1/2} - u^{-3/2}\).

• Find the antiderivative of each term separately.
• For \(u^{-1/2}\), the antiderivative is \(2u^{1/2}\).
• For \(u^{-3/2}\), it is \(-\frac{2}{3}u^{-1/2}\).

After finding the antiderivative, substitute the upper and lower limits. The result is the evaluation of the definite integral between \(2\) and \(e^2+1\), leading to the problem's final solution. Understanding antiderivative calculation allows you to effectively solve definite integrals.