Problem 36 - 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: Understand the Power Rule for Integration

The power rule for integration states that the integral of \(t^n\) with respect to \(t\) is \(\frac{t^{n+1}}{n+1} + C\), where \(n eq -1\) and \(C\) is the constant of integration. This rule allows us to find the antiderivative of power functions.

Step 2: Apply the Power Rule

Given the integral \( \int t^{12} \, dt \), apply the power rule for integration. Set \(n = 12\). Substitute this into the rule to get \( \frac{t^{12+1}}{12+1} + C\).

Step 3: Simplify the Expression

Simplify the expression from Step 2. Calculate \(12+1\) to get \(13\). Therefore, the antiderivative of \(t^{12}\) is \( \frac{t^{13}}{13} + C\).

Step 4: Write the Final Answer

The indefinite integral of \( t^{12} \) with respect to \( t \) is written as \( \frac{t^{13}}{13} + C \). Include the constant of integration \( C \) at the end.

Key Concepts

Power Rule for Integration

The power rule for integration is a fundamental technique in calculus used to integrate polynomial functions. It's especially helpful because it simplifies the process of finding integrals significantly. Given a function of the form \( t^n \), the power rule provides a straightforward way to find its integral.

• The rule states that \( \int t^n \, dt = \frac{t^{n+1}}{n+1} + C \), where \( n eq -1 \).
• This formula increases the exponent \( n \) by 1 (hence \( t^{n+1} \)), and then divides the result by the new exponent \( n+1 \).
• "C" represents a constant of integration that accounts for any constant term that could be differentiated to zero.

Using this rule, integrals of polynomials become more manageable. Always ensure that \( n \) in the expression \( t^n \) is not -1, as this would require a different method of integration.

Antiderivative

An antiderivative, also known as an indefinite integral, is a function that reverses the process of differentiation. When we seek an antiderivative, we're essentially looking for a function whose derivative gives back the original function.

• In the context of our exercise, the antiderivative of \( t^{12} \) is \( \frac{t^{13}}{13} + C \).
• This is because when \( \frac{t^{13}}{13} \) is differentiated, it returns \( t^{12} \).
• Every derivative operation has multiple possible antiderivatives differing by a constant, which is where "C" comes in.

Antiderivatives are crucial because they allow us to undo differentiation and find original functions from derivative expressions. They are widely used in calculating areas under curves among other applications.

Constant of Integration

The constant of integration, represented by "C", is an essential part of indefinite integrals. This constant is added because differentiating a constant yields zero, making it invisible in the derivation process.

• Any function whose derivative is zero can be part of the integral solution; hence, "C" ensures all potential solutions are accounted for.
• When evaluating indefinite integrals, the constant of integration is crucial to capture the entire family of potential antiderivatives.
• In our example, the indefinite integral \( \int t^{12} \, dt = \frac{t^{13}}{13} + C \) includes "C" to signal the range of possible original functions.

Remembering to include the constant of integration is key in calculus. It signifies the most general form of the antiderivative, encompassing all potential shifts or translations of the integrated function.