Problem 11 - Calculus and its applications

Question

Find each integral. $$ \int \frac{1}{x^{3}} d x $$

Step-by-Step Solution

Answer

The integral is \( -\frac{1}{2x^{2}} + C \).

Step 1: Identify the form of the integrand

Recognize that the integrand is in the form of a power of x, specifically \( x^{-3} \). We will use the power rule for integration, which states \( \int x^{n} \, dx = \frac{x^{n+1}}{n+1} + C \), where \( n eq -1 \).

Step 2: Apply the power rule for integration

Rewrite the integrand \( \frac{1}{x^3} \) as \( x^{-3} \) and apply the power rule, giving us \( \int x^{-3} \, dx = \frac{x^{-3+1}}{-3+1} + C = \frac{x^{-2}}{-2} + C \).

Step 3: Simplify the result

Simplify the expression found in Step 2. We get \( \frac{x^{-2}}{-2} + C = -\frac{1}{2x^{2}} + C \), where \( C \) is the constant of integration due to the indefinite integral.

Key Concepts

Power Rule

The Power Rule is a fundamental concept in integration, used for finding the integral of a power of a variable. In simpler terms, it helps us manage expressions like \( x^n \). The rule itself states:

• \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \)

where \( n eq -1 \), since dividing by zero is undefined.
This straightforward method allows us to increment the exponent of \( x \) by 1 and then divide by the new exponent. Don't forget the \( C \) at the end—it's essential!
For this specific exercise, the function initially given as \( \frac{1}{x^{3}} \) was converted to \( x^{-3} \). Applying the Power Rule delivers:

• \( \int x^{-3} \, dx = \frac{x^{-2}}{-2} + C \)

which gives us the simplified final result.

Definite and Indefinite Integrals

Integrals can be either definite or indefinite.

• Indefinite integrals include the constant of integration, \( C \), and no numerical limits.
• Definite integrals have specific upper and lower bounds and evaluate to a numerical value.

For indefinite integrals, like in this exercise, you indicate the anti-derivative of the function, which includes an arbitrary constant \( C \). This is due to the fact that integration, essentially the reverse of differentiation, always involves a family of functions shifted vertically.
Unlike indefinite integrals, definite integrals compute the area under a curve within specific boundaries, and the solution is a specific numeric value rather than a function.

Constant of Integration

When dealing with indefinite integrals, always append \( C \), referred to as the constant of integration.

• Why \( C \) is vital: It accounts for all potential vertical shifts in the graph of the antiderivative.
• The equation \( y = f(x) + C \) represents a family of functions differing only in their constant \( C \).

Every time we take an indefinite integral, there is not just one function but infinitely many functions that differ by a constant. The constant of integration ensures we consider all these potential solutions.
In this exercise, during the integration of \( x^{-3} \), the \( C \) demonstrates these multiple possibilities, as it represents all individual solutions differing only by this constant factor.