Problem 382 - Calculus Volume 1

Question

Show that if \(c>0\), then the integral of \(1 / x\) from \(a c\) to \(b c(0

Step-by-Step Solution

Answer

Both integrals equal \( \ln\left( \frac{b}{a} \right) \).

Step 1: Set Up the Integral with Substitution

We are tasked to show that \( \int_{ac}^{bc} \frac{1}{x} \, dx = \int_a^b \frac{1}{x} \, dx \). To solve this, we perform a substitution with \( u = \frac{x}{c} \), so \( x = uc \) and \( dx = c \, du \). The integration bounds change from \( x = ac \) to \( u = a \) and from \( x = bc \) to \( u = b \).

Step 2: Substitute and Change Limits

Substituting these into the integral from Step 1 gives: \[ \int_{ac}^{bc} \frac{1}{x} \, dx = \int_a^b \frac{1}{uc} \, c \, du \]. This simplifies to \( \int_a^b \frac{1}{u} \, du \) because \( c \) and \( \frac{1}{c} \) cancel each other out.

Step 3: Find the Simplified Integral

The integral simplification from Step 2 is the same as the right-hand side of the original problem statement, which is \( \int_a^b \frac{1}{x} \, dx \). Integrating \( \frac{1}{u} \) gives \( \ln|u| \), so the integral evaluates to: \[ \ln|b| - \ln|a| = \ln\left( \frac{b}{a} \right) \]. Both the original and substituted integrals evaluate to the same expression.

Step 4: Conclude and Verify

Since both integrals, \( \int_{ac}^{bc} \) and \( \int_a^b \), simplify to \( \ln\left( \frac{b}{a} \right) \), we can conclude that the integrals are equal regardless of the chosen \( c \) value greater than zero. Thus, we've verified the problem statement is true.

Key Concepts

change of variables

The change of variables is an essential technique in integral calculus. It's particularly useful for simplifying integrals by transforming them into a more manageable form. It involves substituting the original variable of integration with a new one, which leads to a change in the limits of integration as well.
This allows for the integral to be rewritten in terms of the new variable. For example, if we have an integral involving the variable \( x \), we might substitute \( x = u \cdot c \) (where \( u \) is the new variable and \( c \) is a constant) to simplify things.
This substitution also requires us to modify the differential appropriately, meaning we must replace \( dx \) with \( c \, du \), since the differential \( dx \) relates to the rate of change of \( x \) with respect to \( u \). The change of variables often results in the cancellation of constants, making the integration process smoother.

definite integrals

Definite integrals are used to calculate the area under a curve within specified limits. They require not only the function to be integrated but also defined limits over which integration takes place. This means they are specified from point \( a \) to point \( b \).
The result of a definite integral provides a numeric value representing the area, unlike an indefinite integral, which results in a general function plus a constant.
Definite integrals are fundamental in finding quantities like area, volume, and displacement, where specific bounds or limits are applied. A property of definite integrals is their dependence on the integration limits. When these limits are transformed during substitution, they need to be updated accordingly to reflect the new variable of integration.

• They have clear limits for integration.
• Provide a specific numerical value as a result.
• Require careful adjustment of limits when changing variables.

In the exercise, we see the definite integral from \( ac \) to \( bc \) transformed to the bounds \( a \) to \( b \).

substitution method

The substitution method is a powerful technique used to simplify integrations. It's particularly useful when dealing with complicated integrands that can't be easily integrated in their current form. By substituting part of the integral with a new variable, we can sometimes transform the integral into a simpler one.
The substitution method involves a few critical steps:

• Choose a new variable to substitute in place of part of the original integrand.
• Replace the original variable with the new one, adjusting both the differential \( dx \) and the integration limits.
• Simplify and solve the resulting integral.

In the exercise, we used the substitution \( u = \frac{x}{c} \) to transform the given integral. This simplification reduced it to the form \( \int_a^b \frac{1}{u} \, du \), which is simpler and follows naturally from basic integration rules.
By simplifying the expression through substitution, we could demonstrate that both original and transformed integrals evaluate to the same value. Thus, the substitution method enables us to tackle otherwise complex integrals with greater ease and accuracy.