91Ó°ÊÓ

Integration by substitution is a powerful technique used to simplify complex integrals. It involves changing variables to transform the integral into a simpler form that is easier to solve. For instance, if given an integrand with a composite function, such as an expression involving exponents and square roots, substitution can be very helpful.

To use integration by substitution, follow these steps:

• Identify a substitution that will simplify the integral. This is often a part of the integrand that appears more complex, like the expression inside a square root or an exponential function.
• Express the substituted variable in terms of the original variable. For example, if you choose a substitution like \(u = 3 + 4e^t\), express \(du\) in terms of \(dt\).
• Substitute \(u\) and \(du\) into the integral to complete the transformation.

This technique effectively transforms an integral into a form that matches standard integrals known from common reference tables, simplifying the overall process.

A table of integrals is a handy reference tool for solving indefinite integrals. It lists common integral forms and their solutions, saving you time and effort during calculations. Tables usually cover standard functions like power functions, exponential functions, and trigonometric functions.

Once you have transformed an integral using substitution, compare it to known forms in your table. In the exercise, for example, the integral \(\int \frac{1}{\sqrt{u}} du\) is recognized as a standard form that integrates to \(2\sqrt{u} + C\).

• Use the table to directly find the antiderivative.
• Consider constants outside the integral to adjust the final solution, like multiplying by \(\frac{1}{4}\) in this case, which results from the substitution.

Tables of integrals are excellent resources when dealing with complex expressions, allowing you to bypass time-consuming algebra and focus on interpretation and application of solutions.

The change of variables is a concept closely related to integration by substitution. It includes redesigning the function's variables to simplify the integration process. The idea is to convert complex integrals into easier forms by introducing new variables.

When performing a change of variables:

• Decide on a new variable and express other parts of the integral through it, such as setting \(u = 3 + 4e^t\).
• Calculate the differential of the new variable, here \(du = 4e^t dt\), and replace \(dt\) in the integral accordingly.
• Reformulate the integral using these new terms, making it simpler to evaluate.

After solving the integral in terms of the new variable, always remember to substitute the original variables back into the expression, yielding an intuitive solution in terms of the initial question components. This technique is invaluable for handling integrals of complicated expressions.