91Ó°ÊÓ

In calculus, the substitution method is a technique used to simplify an integral by making a substitution that changes the variable of integration. This often transforms the integral into a simpler form that is easier to solve. In the given exercise, we encounter a composite function which makes direct integration complicated. By identifying a part of the function as a new variable, we can reduce the complexity of the integral.

For instance, with the integral \( \int e^{x} \sqrt{1+e^{x}} \ dx \), we choose the substitution \( u = 1 + e^x \). This choice is strategic because it directly relates to the composite function \( \sqrt{1+e^{x}} \). This substitution transforms the integral into \( \int \sqrt{u} \ du \), which is much simpler to integrate.

Using substitution:

• Identify a portion of the integral to represent \( u \).
• Calculate \( du \) in terms of the original variable.
• Substitute \( u \) and \( du \) into the integral.

This method is very helpful for dealing with integrals involving composite functions or when the integral resembles a known derivative.

A composite function is a function made from two or more functions in such a way that the output of one function becomes the input of the next. In this context, we are dealing with \( e^{x} \sqrt{1+e^{x}} \) as a composite function.

Understanding composite functions:

• Function composition is characterized by nested functions.
• The inner function \( g(x) \) is placed inside the outer function \( f(g(x)) \).
• The derivative or integral of a composite function often involves using the chain rule or substitution method.

In integrals like \( \int e^{x} \sqrt{1+e^{x}} \ dx \), noticing the composite nature signals the potential need for substitution. Here, setting \( u = 1 + e^x \) simplifies the evaluation of the integral by isolating the complexity brought on by the interaction of \( e^x \) and \( \sqrt{1+e^{x}} \). Recognizing composite functions quickly becomes a helpful skill in calculus for streamlining solutions.

The power rule for integration is a basic technique that enables us to integrate functions of the form \( x^n \). This rule is especially useful when simplified expressions are obtained during integration, such as when using substitution to transform complex expressions.

The power rule states:
\[ \int x^n \ dx = \frac{x^{n+1}}{n+1} + C \]
where \( C \) is the constant of integration, and \( n eq -1 \).

Applying this can turn an unwieldy integral into a straightforward calculation. When solving the integral \( \int \sqrt{u} \, du \), recognize that \( \sqrt{u} = u^{1/2} \). Using the power rule:

• Identify \( n \), which in this case is \( 1/2 \).
• Apply the formula: \( \frac{u^{3/2}}{3/2} + C \).
• Simplify to \( \frac{2}{3} u^{3/2} + C \).

This shows how effectively the power rule breaks down integrals once they are in the suitable form, highlighting why initial problem simplification is so beneficial.