Problem 17 - Applied Calculus

Question

Find each indefinite integral. \(\int \frac{3}{2 v} d v\)

Step-by-Step Solution

Answer

\( \frac{3}{2} \ln |v| + C \)

Step 1: Identify the Integral Form

Recognize that the integral given is of the form \( \int \frac{a}{x} \, dx \), which can be directly integrated using the natural logarithm rule.

Step 2: Apply the Natural Logarithm Rule

The integral \( \int \frac{1}{x} \, dx \) is equal to \( \ln |x| + C \), where \( C \) is the constant of integration. For the given integral, \( a = \frac{3}{2} \), so it becomes \( \frac{3}{2} \int \frac{1}{v} \, dv \).

Step 3: Integrate Using the Coefficient Rule

Factor out the constant coefficient \( \frac{3}{2} \) from the integral to write it as \( \frac{3}{2} \ln |v| + C \). This is the result of integrating \( \int \frac{1}{v} \, dv \).

Step 4: Write the Solution

Thus, the indefinite integral \( \int \frac{3}{2v} \, dv \) is \( \frac{3}{2} \ln |v| + C \), where \( C \) is the integration constant.

Key Concepts

Indefinite Integrals

When we talk about indefinite integrals, we are essentially dealing with a function whose derivative is given. Unlike definite integrals, which calculate an area or a value over an interval, indefinite integrals focus on the general form of antiderivatives.

• An indefinite integral is represented by the integral symbol followed by a function and its differential, such as \( \int f(x) \, dx \).
• The result is a family of functions, known as antiderivatives, which differ only by a constant.

This is crucial because the process reverses differentiation, leading us back to the original function (up to a constant). To solve the exercise, we integrate the expression \( \frac{3}{2v} \) without specifying boundaries, so the result will always include \( +C \), representing any constant number.

Natural Logarithm Rule

One of the most handy rules in calculus is the natural logarithm rule. It simplifies the integration process significantly when applied correctly. This rule comes in when you have an integral of the form \( \int \frac{1}{x} \, dx \). The result is the natural logarithm of the absolute value of \( x \), which is expressed as \( \ln |x| \).

• The natural logarithm rule is applied because taking the derivative of \( \ln |x| \) returns \( \frac{1}{x} \), which perfectly aligns with the original integral we are trying to solve.
• In the exercise, identifying the integral formula allows us to use this rule on \( \frac{1}{v} \), resulting in \( \ln |v| \).

Using this rule makes the integration process extremely efficient, cutting down complex algebraic manipulations otherwise necessary.

Constant of Integration

When dealing with indefinite integrals, we must always remember the constant of integration, denoted by \( C \). This term represents an unknown constant that is added to the function after integrating.

• The constant accounts for the fact that when you take the derivative of a constant, it vanishes.
• Thus, any constant in the original function before differentiation can't be recovered through integration without incorporating \( C \).

In our indefinite integral \( \int \frac{3}{2v} \, dv \), the "+ \( C \)" in \( \frac{3}{2} \ln |v| + C \) indicates that there is a whole set of functions that differ only by a constant. This is very important because it tells us that the antiderivative family extends over all these possible constants, showcasing the general nature of the solution.