Problem 45 - Applied Calculus

Question

Find each indefinite integral by the substitution method or state that it cannot be found by our substitution formulas. $$ \int \frac{e^{\sqrt{x}}}{\sqrt{x}} d x \quad[\text { Hint: Let } u=\sqrt{x} .] $$

Step-by-Step Solution

Answer

The integral is \( 2e^{\sqrt{x}} + C \).

Step 1: Choose the Substitution

The hint suggests that we use the substitution \( u = \sqrt{x} \). So, we need to express \( dx \) in terms of \( du \).

Step 2: Differentiate the Substitution

To find \( dx \) in terms of \( du \), differentiate \( u = \sqrt{x} \) with respect to \( x \). We have \( u = x^{1/2} \), so differentiating gives \( du = \frac{1}{2}x^{-1/2} dx = \frac{1}{2\sqrt{x}} dx \). This implies \( dx = 2\sqrt{x} \, du \).

Step 3: Substitute Variables in the Integral

Substitute \( u = \sqrt{x} \) and \( dx = 2\sqrt{x} \, du \) into the integral: \( \int \frac{e^{\sqrt{x}}}{\sqrt{x}} \, dx \) becomes \( \int \frac{e^u}{u} \cdot u \cdot 2 \, du \). This simplifies to \( 2 \int e^u \, du \).

Step 4: Evaluate the New Integral

The integral \( \int e^u \, du \) is a basic exponential integral, resulting in \( e^u + C \), where \( C \) is the constant of integration. Therefore, \( 2 \int e^u \, du = 2(e^u + C) = 2e^u + C' \), where \( C' = 2C \).

Step 5: Back-Substitute to Original Variable

Convert back to the original variable using \( u = \sqrt{x} \). So, \( 2e^u + C' = 2e^{\sqrt{x}} + C' \).

Key Concepts

Substitution Method

The Substitution Method is a powerful tool for finding indefinite integrals, especially when the integral seems complex at first glance. The goal is to simplify the integral by changing variables. This involves substituting a new variable, often denoted as "u," in place of a more complicated expression.
For example, in the given problem, the hint suggests using the substitution \( u = \sqrt{x} \). This helps to simplify the integrand, making the integration process more manageable.

• Identify a part of the integrand that can be replaced by a new variable.
• Express the original differential \( dx \) in terms of the new variable \( du \).
• Rewrite the integral with the new variable and integrate using known formulas.

The substitution method greatly simplifies the process of dealing with otherwise difficult integrals, turning them into more familiar forms.

Differentiation

Differentiation is the process of finding the derivative of a function, which measures how a function changes as its input changes. In the context of substitution in integrals, differentiation is used to find the relationship between the original variable and the new variable.
In our example, after substituting \( u = \sqrt{x} \), we need to determine how \( dx \) relates to \( du \). This involves differentiating \( u = x^{1/2} \) with respect to \( x \), yielding \( du = \frac{1}{2}x^{-1/2}dx = \frac{1}{2\sqrt{x}}dx \).

• Differentiation helps convert from one variable to another within an integral.
• The derivative provides a means to express \( dx \) in terms of the new variable \( du \).

By understanding differentiation in this context, we can effectively transform and solve integrals using substitution.

Exponential Integral

An exponential integral involves integrating expressions containing the exponential function \( e^x \). These are among the most straightforward integrals because the integral of \( e^u \) with respect to \( u \) is simply \( e^u + C \), where \( C \) is the constant of integration.
In the given problem, once we've made the substitution \( u = \sqrt{x} \), the resulting integral is \( 2 \int e^u \, du \). This directly results in \( 2(e^u + C) \), or \( 2e^u + C' \).
This straightforward nature of exponential integrals simplifies the solution process, providing a clear and concise way to handle integrals where the exponential function is present.

Constant of Integration

In indefinite integrals, the constant of integration \( C \) represents an unknown constant added to the antiderivative. It arises because derivatives of constant terms are zero, so when we integrate, we must account for all possible constants.
In our example, after evaluating \( 2 \int e^u du \), we introduce a constant of integration as \( C' = 2C \), which reflects the fact that any constant can be added to \( e^u \) and still be a solution.

• Ensure that a constant of integration is always included in indefinite integrals.
• The constant can be adjusted based on any multiplying factors from the integral.

This step is essential to ensure the generality and completeness of your integral solutions.