Problem 54 - Applied Calculus

Question

Find the indefinite integrals. $$\int t^{12} d t$$

Step-by-Step Solution

Answer

\( \int t^{12} \, dt = \frac{t^{13}}{13} + C \).

Step 1: Recognize the Problem

We are tasked with finding the indefinite integral \( \int t^{12} \, dt \). This is a typical case of applying the power rule for integration.

Step 2: Apply the Power Rule for Integration

When integrating \( t^n \) with respect to \( t \), where \( n eq -1 \), we use the power rule formula: \( \int t^n \, dt = \frac{t^{n+1}}{n+1} + C \).

Step 3: Substitute the Given Exponent

In our problem, the exponent \( n \) is 12. Substitute \( n = 12 \) into the formula: \( \int t^{12} \, dt = \frac{t^{12+1}}{12+1} + C \).

Step 4: Simplify the Expression

Calculate \( 12 + 1 = 13 \), so the integral becomes \( \frac{t^{13}}{13} + C \). There are no like terms to combine here, so this is the final expression.

Key Concepts

Power Rule for Integration

The power rule for integration is a vital tool to solve integrals involving terms raised to a power. Since integrals are essentially the reverse process of differentiation, understanding this concept can help you solve a wide range of problems in calculus.
When dealing with integrals like \( \int t^n \, dt \), where \( n \) is any real number except \( -1 \), the power rule comes in handy. The rule states that:

• \( \int t^n \, dt = \frac{t^{n+1}}{n+1} + C \)

Here, \( n+1 \) indicates that we increase the exponent by one.
The division by this new exponent \( (n+1) \) adjusts the coefficient of the term to account for this increase.
Lastly, we add \( C \), the constant of integration, to represent any constant that could be added, since the derivative of a constant is zero.

Step by Step Solution

Breaking down problems step by step makes complex calculations much simpler to grasp. Here's the approach to solving the integral \( \int t^{12} \, dt \) using our previous understanding of the power rule for integration:
First, note that the goal is to find an antiderivative for the term \( t^{12} \). This means we're looking for a function whose derivative is \( t^{12} \). By identifying this need, we determine the power rule is appropriate.

• Identify the exponent: Here, \( n = 12 \).
• Apply the power rule: Substituting \( n = 12 \) into our power rule formula gives us \( \frac{t^{12+1}}{12+1} + C \).
• Calculate the new exponent: \(12 + 1 = 13\), so the expression becomes \( \frac{t^{13}}{13} + C \).
• Present the final solution: Since no terms need combining, the simplified form \( \frac{t^{13}}{13} + C \) is our answer.

Make sure to practice similar problems to get fully comfortable with this approach as repetition strengthens understanding.

Integrals in Calculus

Integrals are a fundamental concept in calculus, serving as the opposite operation to differentiation. They're primarily used to calculate areas under curves, among many other applications in mathematics and science.
There are two main types of integrals:

• Definite Integrals: These calculate the accumulated area under a curve between two points and result in a numerical value.
• Indefinite Integrals: These provide a general form solution, representing a family of functions. They include a constant \( C \), as any constant could have been added after differentiating a solution to yield the same original function.

Understanding the process of integration and the interpretation of its results allows you to solve various real-world and theoretical problems.
Integrals also establish the foundation for advanced topics like differential equations and multivariable calculus, making their mastery essential for further studies in mathematics.