Problem 98 - Calculus Single Variable

Question

Determine the value of the upper limit of integration \(b\) for which a substitution converts the integral on the left to the integral on the right. \(\int_{1}^{b}(x+1) \exp (1 / x-\ln (x)) / x^{2} d x=\int_{1 / 4}^{1} \exp (u) d u\)

Step-by-Step Solution

Answer

The upper limit of integration \( b \) is the solution to \( \frac{1}{b} - \ln(b) = \frac{1}{4} \).

Step 1: Identify the substitution

Notice that the given integral on the left can be transformed into the integral on the right using an appropriate substitution. The right side integral has the form \( \int \exp(u) \, du \), so let's define the substitution \( u = \frac{1}{x} - \ln(x) \).

Step 2: Compute the derivative

To find \( du \) in terms of \( x \), calculate the derivative of \( u = \frac{1}{x} - \ln(x) \). The derivative is \( du = -\frac{1}{x^2} - \frac{1}{x} \, dx = -\left( \frac{1+x}{x^2} \right) \, dx \).

Step 3: Adjust the integral

Substitute into the integral: \( \int (x+1) \frac{\exp(1/x - \ln{x})}{x^2} \, dx \). Replace \( (x+1)/x^2 \, dx \) with \(-du \) based on the derivative computed in Step 2.

Step 4: Adjust limits of integration

Determine the new limits for \( u \) when \( x = 1 \), \( u = \frac{1}{1} - \ln(1) = 1 \). When \( x = b \), \( u(b) = \frac{1}{b} - \ln(b) \). Set these equal to the right integral's limits: from \( \frac{1}{4} \) to \(1\).

Step 5: Solve for b

Set \( \frac{1}{b} - \ln(b) = \frac{1}{4} \). Solve this equation for \( b \) to find the value of the upper limit of integration that makes the substitution work.

Key Concepts

Substitution Method

The substitution method is a powerful tool used in integral calculus to transform complex integrals into simpler ones. It involves redefining the variable of integration to simplify the mathematical expression. In our original exercise, the substitution chosen is \( u = \frac{1}{x} - \ln(x) \). This transformation helps align the given integral with a standard integral form, making it easier to solve.

• Substitution is akin to changing the perspective of the problem, allowing us to look at it in a new light.
• This method requires calculating the derivative \( du \) in terms of the original variable \( x \).
• Adjust the integral by replacing old variable terms with the new ones, as given by the derivative.
• Finally, fold this new substitution back into a simpler integral format, which in this case is \( \int e^u \, du \).

With a proper substitution, we achieve a much simpler integral that's aligned with standard forms, significantly easing the computation process.

Definite Integrals

Definite integrals calculate the area under a curve from one point to another along the x-axis. In contrast to indefinite integrals, definite integrals have specific upper and lower limits of integration.

• These limits define the range over which the function is being calculated.
• Definite integrals result in a numeric value representing this area.
• For example, \( \int_{1}^{b} (x+1) e^{(1/x - \ln(x))}/x^2 \, dx \) has limits from 1 to \( b \).

Using substitution might change the limits, so these must be correctly adjusted to reflect the new variable. This adjustment is crucial to ensure the areas covered remain consistent before and after substitution. The ultimate goal is to compute the exact value of the definite integral accurately.

Limits of Integration

In integration, the limits of integration refer to the values that define the endpoint values of the integral—both start (lower limit) and stop (upper limit). When performing a substitution, these limits often need to be transformed as well.

• In this exercise, the limits of the integral change due to the substitution \( u = \frac{1}{x} - \ln(x) \).
• When \( x = 1 \), the lower limit becomes \( u = 1 \).
• For the upper limit, solving \( \frac{1}{b} - \ln(b) = \frac{1}{4} \) finds the value of \( b \) that matches the endpoint.

Such calculations ensure the integrity of the integral is maintained so that it accurately represents the area or value intended. The right change allows us to perform substitutions without altering the initial conditions' alignment.