Problem 23 - Calculus Early Transcendentals

Question

Evaluate the following integrals. $$\int \frac{e^{\sqrt{x}}}{\sqrt{x}} d x$$

Step-by-Step Solution

Answer

Question: Evaluate the integral \(\int \frac{e^{\sqrt{x}}}{\sqrt{x}} dx\). Answer: The solution of the given integral is \(\int \frac{e^{\sqrt{x}}}{\sqrt{x}} dx = e^{2\sqrt{x}} + C\), where \(C\) is the constant of integration.

Step 1: Substitution

Substitute \(u = e^{\sqrt{x}}\), find the derivative of \(u\) w.r.t. \(x\) and substitute. $$\frac{d u}{d x} = e^{\sqrt{x}} \cdot \frac{1}{2\sqrt{x}}$$ $$d u = \frac{e^{\sqrt{x}}}{2\sqrt{x}} d x$$ Now, substitute \(u\) and \(d u\) into the given integral: $$\int \frac{e^{\sqrt{x}}}{\sqrt{x}} d x = \int 2u d u$$

Step 2: Integrate

Now, integrate the new expression with respect to \(u\). $$\int 2u d u = u^2 + C$$

Step 3: Substitute back

Substitute back the value of \(x\) in terms of \(u\). $$u^2 + C = (e^{\sqrt{x}})^2 + C$$ $$e^{2\sqrt{x}} + C$$ So, the solution of the given integral is: $$\int \frac{e^{\sqrt{x}}}{\sqrt{x}} d x = e^{2\sqrt{x}} + C$$

Key Concepts

Substitution method

The substitution method is a powerful technique used in integration when a direct approach is not feasible. The idea is to simplify an integral by substituting a part of it with a new variable, which makes it easier to solve. Usually, this method is applied when an integral contains a composition of functions, or when there's a visible function and its derivative.

• Finding a substitute: Identify a function or part of the integral that resembles another function's derivative. For example, in the integral \( \int \frac{e^{\sqrt{x}}}{\sqrt{x}} dx \), the expression \( e^{\sqrt{x}} \) alongside its derivative is visible.
• Derivative computation: Compute the derivative of the chosen substitution function to facilitate a change in variables. Thus, substituting \( u = e^{\sqrt{x}} \) makes use of the derivative \( \frac{du}{dx} = \frac{e^{\sqrt{x}}}{2\sqrt{x}} \).
• Substitute and transform: Replace the original variables and differential with the new ones. This transforms the integral into a more straightforward form, such as \( \int 2u \, du \), making the integration process easier.

The substitution method simplifies complex integrals and lays the groundwork for a more straightforward integration process.

Exponential functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. They have pivotal importance in calculus, particularly because of their unique derivative and integral properties, which often simplify complex calculations.

• Basic form: The simplest exponential function is \( e^x \), where \( e \) is Euler's number, approximately equal to 2.71828. This function is special because its derivative and integral are the same.
• Properties: Exponential functions grow rapidly, and this growth paradigm is captured in their derivatives and integrals. Specifically, the derivative of \( e^{f(x)} \) is \( f'(x)e^{f(x)} \), and the integral of \( e^{f(x)} \) is typically \( rac{1}{f'(x)}e^{f(x)} \), assuming \( f(x) \) is a suitable function.

In calculus, exponential functions facilitate solving integrals involving growth and decay models, such as population growth or radioactive decay, due to their inherent property of derivative equivalence.

Indefinite integrals

Indefinite integrals represent an entire family of functions whose derivatives are the given function in the integral. They are useful for finding the antiderivative of a function, and unlike definite integrals, they do not evaluate to a specific number.

• General form: The indefinite integral of a function \( f(x) \) is denoted as \( \int f(x) \, dx \), and its result is \( F(x) + C \), where \( C \) is the integration constant.
• Constant of integration: The constant \( C \) reflects the family of solutions, since an infinite number of functions differ only by a constant can have the same derivative.

In the integral \( \int \frac{e^{\sqrt{x}}}{\sqrt{x}} dx \), the indefinite integral results in \( e^{2\sqrt{x}} + C \), indicating the general solution to the integral without assigning specific boundary values.