Problem 8 - Applied Calculus

Question

Find each indefinite integral. \(\int \frac{d w}{w^{2}}\)

Step-by-Step Solution

Answer

\( \int \frac{d w}{w^{2}} = -\frac{1}{w} + C \)

Step 1: Identify the Integral Type

Notice that the integral \( \int \frac{d w}{w^{2}} \) contains a power of \( w \) in the denominator. This can be rewritten as \( \int w^{-2} \, dw \) to identify it as a power rule integral.

Step 2: Apply the Power Rule

The power rule for integration is \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( n eq -1 \). In our case, \( n = -2 \). Applying the rule gives: \( \int w^{-2} \, dw = \frac{w^{-2+1}}{-2+1} + C = \frac{w^{-1}}{-1} + C \).

Step 3: Simplify the Expression

The expression \( \frac{w^{-1}}{-1} \) simplifies to \( -w^{-1} \). Further, \( w^{-1} \) can be rewritten as \( \frac{1}{w} \). So the integral becomes \( -\frac{1}{w} + C \).

Step 4: Write the Final Answer

Thus, the indefinite integral \( \int \frac{d w}{w^{2}} \) evaluates to \( -\frac{1}{w} + C \). Here, \( C \) is the constant of integration.

Key Concepts

Power Rule for Integration

The power rule for integration is a cornerstone concept in calculus that simplifies the process of finding indefinite integrals. When we talk about indefinite integrals, we are looking to reverse the process of differentiation. The power rule specifically applies to functions of the form \( x^n \), where \( n \) is any real number except \(-1\). The rule is stated as:

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

To apply the power rule,

• Increment the exponent \( n \) by 1.
• Divide the result by the new exponent.
• Add the constant of integration \( C \).

In the exercise example, the integral \( \int w^{-2} \, dw \) is neatly solved using the power rule:

• \( n = -2 \). Incrementing by 1 gives \( n+1 = -1 \).
• The result is \( \frac{w^{-1}}{-1} = -w^{-1} \).
• Finally, convert \( w^{-1} \) back to \( \frac{1}{w} \), which provides the integral as \( -\frac{1}{w} + C \).

Constant of Integration

The constant of integration, often denoted as \( C \), is a crucial aspect of indefinite integrals, yet it can sometimes be overlooked. When we integrate a function indefinitely, we're essentially finding all possible antiderivatives.
Because derivatives eliminate constant terms (since a constant's derivative is zero), there's no unique "parent function" for an antiderivative without specifying \( C \).- Therefore, \( C \) accounts for any constant that might have been present in the original function before differentiation.For instance, if you differentiate \( f(x) = x^2 + 5 \), the result is \( 2x \), but if you integrate \( 2x \), your result could be \( x^2 \), \( x^2 + 5 \), or \( x^2 - 3 \), and so on. That's why we add \( C \) in indefinite integrals, to cover all these possibilities. In the sample exercise, even though we found \( -\frac{1}{w} \), the indefinite nature requires us to include \( + C \) to indicate there could be other such functions.

Integral Notation

Integral notation is an integral (pun intended!) part of understanding calculus expressions and operations. It helps communicate the concept of integration efficiently and precisely.

• The integral sign \( \int \) originates from the long "s" used in the Latin word "summa," meaning "sum." It implies a summation process happening throughout the range of integration.
• The expression after \( \int \) within the integrand, followed by the differential (like \( dw \)), indicates the variable of integration.

In an indefinite integral like \( \int w^{-2} \, dw \), we aren't concerned with specific bounds; rather, the integral represents a family of functions that derive to produce the given integrand. Integral notation is versatile, accommodating polynomials, trigonometric functions, and more complex expressions, making it broadly applicable across mathematical contexts.
Correctly reading and interpreting this notation allows you to apply the appropriate rules (like the power rule) to solve for integrals or antiderivatives.