91Ó°ÊÓ

The **substitution method** is a powerful technique used in integration to simplify the process, especially when dealing with composite functions. Think of it as a strategy to change the variable of integration to make the integral more manageable. Essentially, you identify a part of the integrand (the function being integrated) to substitute with a new variable. This substitution should transform the integral into a simpler form.

Here's how it works, step-by-step:

• Identify a substitution: Look for a function within your integral that can be replaced by a new variable, usually denoted as 'u'.
• Compute the derivative: Differentiate your substitution to find the differential du.
• Rewrite the integral: Replace the identified function and differential in terms of 'u'.

In our example, we let \( u = 2t + 1 \). With this choice, the derivative is \( \frac{du}{dt} = 2 \) which gives \( dt = \frac{du}{2} \). This substitution simplifies the original integral into a more familiar form for integration.

The **power rule for integration** is a fundamental concept in calculus that provides a straightforward way to find the antiderivative of a power function. When dealing with integrals that contain simple powers of variables, this rule becomes invaluable:

• The rule states: \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( n eq -1 \).
• The constant \( C \) is added to represent the indefinite nature of antiderivatives.
• Note that the exponent of \( x \) increases by one, and the coefficient divides by the new exponent.

In our problem, after substituting, we were left with \( \int u^{-1/2} \, du \). Applying the power rule here, where \( n = -\frac{1}{2} \), results in \( \frac{u^{1/2}}{1/2} + C \). This step is crucial as it transforms an otherwise complex integral into a solvable expression.

While our exercise resulted in an **indefinite integral**, understanding the definite integral is equally important. A definite integral calculates the net area under a curve between two bounds, providing a numeric result rather than a function.

• Written as \( \int_a^b f(x) \, dx \), it computes the area from \( x = a \) to \( x = b \).
• It requires the use of the Fundamental Theorem of Calculus: the difference between the antiderivative evaluated at the upper bound and the lower bound.
• The result of a definite integral is a specific number rather than a general expression.

Despite not being directly addressed in our solution, understanding definite integrals helps relate antiderivatives to areas, offering insight into practical applications in fields such as physics and engineering.

In contrast, the **indefinite integral** involves finding a function's general antiderivative without specified limits. It represents a family of functions that differ only by a constant, given by \( C \).

• Expressed as \( \int f(x) \, dx \), the result includes the constant of integration, \( C \).
• This constant reflects all potential vertical shifts of the original function's antiderivative.
• Indefinite integrals are key in solving initial value problems within differential equations.

Our exercise's final result, \( 5\sqrt{2t + 1} + C \), is an indefinite integral. The inclusion of \( C \) indicates our solution encompasses all vertical shifts of the resulting antiderivative, making it a universal solution until specific boundary conditions define \( C \).