Problem 12 - Applied Calculus

Question

Find each indefinite integral. \(\int 9 x^{8} d x\)

Step-by-Step Solution

Answer

The indefinite integral is \(x^9 + C\).

Step 1: Identify the Integral Form

The integral given is \(\int 9x^8 \, dx\). This is a simple polynomial integral in the form \(\int ax^n \, dx\) where \(a = 9\) and \(n = 8\).

Step 2: Apply the Power Rule for Integration

Using the power rule for integration, which states \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\), apply it to the integral where \(n = 8\). This becomes: \(\frac{x^{8+1}}{8+1}\) which simplifies to \(\frac{x^9}{9}\).

Step 3: Calculate the Constant Coefficient

Multiply the result by the constant coefficient from the integral, which is 9. Thus, \(9 \cdot \frac{x^9}{9} = x^9\).

Step 4: Add the Constant of Integration

Finally, don't forget to add the constant of integration \(C\) to the result since this is an indefinite integral. So, the solution to the integral is \(x^9 + C\).

Key Concepts

Power Rule for Integration

The power rule for integration is a fundamental tool when dealing with polynomial functions. It's quite similar to the power rule in differentiation, but used in reverse.
This rule tells us how to integrate functions of the form \( x^n \). Specifically, the power rule states:

• If \( n eq -1 \), then \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \).

This simple formula helps us find the indefinite integral of power functions.
Here, \( C \) represents the constant of integration which accounts for all possible vertical shifts of the function on a graph.
In our given example \( \int 9x^8 \, dx \), the power is 8. To apply the rule:

• Increase the power by 1, turning it from \( x^8 \) to \( x^9 \).
• Divide by the new exponent to get \( \frac{x^9}{9} \).

This illustrates how the power rule simplifies the integration process.

Polynomial Integral

Polynomial integrals refer to the integration of polynomial functions, which are expressions consisting of variables raised to whole number powers and their coefficients. These types of functions are straightforward to integrate, especially with the power rule.
A polynomial integral can often be identified by its form, typically expressed as \( \int (ax^n + bx^{n-1} + \, \ldots \, + mx + c) \, dx \).

• Each term is integrated separately.
• The same power rule applies to each term.

In the case of our example, \( \int 9x^8 \, dx \) is a simple polynomial integral where the function \( 9x^8 \) requires applying the power rule once.
After finding the antiderivative, remember to add the constant of integration.

Constant of Integration

In calculus, whenever you find an indefinite integral, it includes a constant of integration, denoted by \( C \). But why is it necessary?
Integrals can represent families of functions. Since differentiation of a constant is zero, when you find an antiderivative without considering the constant, multiple original functions could have been differentiated to give you the same result.
Adding the constant \( C \) ensures that all possibilities of antiderivatives are covered.
For example, consider your final result from an integral process \( x^9 \). Without the \( C \), it only represents one of many potential functions that could differentiate to the integrand.
Thus, by writing \( x^9 + C \), you indicate that any constant shift of this curve is also valid as an antiderivative.