Problem 14 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

In Exercises 1 through 20 , find the indicated indefinite integral. \(\int(t-5)^{12} d t\)

Step-by-Step Solution

Answer

\(\int (t-5)^{12} \, d t = \frac{(t-5)^{13}}{13} + C\)

Step 1 - Identify the integral form

The given integral is \(\textstyle \int (t-5)^{12} \, d t\), which is in the form of \(\int (u)^n \, du\). Here, \((t-5)\) can be treated as \(u\).

Step 2 - Perform a substitution

Let \(u = t - 5\). Therefore, \(du = dt\). This substitution transforms the integral into \(\int u^{12} \, du\).

Step 3 - Integrate using the power rule

The power rule for integration states that \(\int u^n \, du = \frac{u^{n+1}}{n+1} + C\). Applying this rule to \(\int u^{12} \, du\), we get \(\int u^{12} \, du = \frac{u^{13}}{13} + C\).

Step 4 - Substitute back to the original variable

Replace \(u\) with \(t - 5\) to revert to the original variable. Thus, \(\frac{u^{13}}{13} + C\) becomes \(\frac{(t-5)^{13}}{13} + C\).

Key Concepts

Integration by Substitution

Integration by substitution is a fundamental method used to simplify and solve integrals. Essentially, it allows us to make complicated integrals easier by substituting parts of the integral with a simpler variable. For example, in the given problem \(\textstyle \textstyle \textstyle \textstyle \int (t-5)^{12} \, d t\), we substitute \(u = t - 5\). This substitution transforms the integral to a simpler form: \(\textstyle \textstyle \textstyle \textstyle \int u^{12} \, du\). The idea is to find a function inside the integral that, when replaced, makes the integration process more straightforward.
Here are the steps to follow:

• Identify a part of the integrand that can be substituted with a new variable (in this case, \(t - 5\)).
• Determine the differential of the new variable (\(du = dt\)).
• Rewrite the integral in terms of the new variable.

By performing these steps, we reduce the complexity of the integral, making it easier to apply standard integration rules.

Power Rule

The power rule is one of the basic rules of integration, which states that \(\textstyle \int u^n \, du = \frac{u^{n+1}}{n+1} + C\). This rule is incredibly useful when dealing with polynomial functions. In the context of our problem, after substituting \(u = t - 5\), we get \(\textstyle \int u^{12} \, du\).
The power rule can be directly applied here:

• Increase the exponent by one (from 12 to 13).
• Divide by the new exponent (13 in this case).
• Don’t forget to add the constant of integration (C) at the end.

Applying the power rule, we find that the integral simplifies to: \(\int u^{12} \, du = \frac{u^{13}}{13} + C\).
This step leverages the simple formula to quickly find the antiderivative.

Calculus Problems

Calculus problems, specifically those involving integration, often look daunting at first. However, by breaking the problem into smaller, manageable steps, as demonstrated in this exercise, you can solve them more efficiently. Understanding methods like substitution and rules like the power rule is crucial.
When faced with calculus problems:

• First, identify the most complex part of the integrand.
• Consider substitutions that can simplify the integral.
• Apply the necessary integration rules to find the antiderivative.

In our problem, these steps allow us to transform and solve \(\textstyle \int (t-5)^{12} \, dt\) into a simpler form, ultimately yielding the solution \(\frac{(t-5)^{13}}{13} + C\). By mastering these techniques, any calculus problem becomes more approachable and manageable.