Problem 96 - Calculus for Biology and Medicine

Question

Compute the indefinite integrals. $$ \int\left(\frac{1}{\sqrt{x}}+\frac{1}{\sqrt{e^{x}}}\right) d x $$

Step-by-Step Solution

Answer

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

Step 1: Simplify the Integral Expression

First, break down the given integral expression into two separate integrals. The expression \( \int \left( \frac{1}{\sqrt{x}} + \frac{1}{\sqrt{e^x}} \right) dx \) can be rewritten as two individual integrals: \( \int \frac{1}{\sqrt{x}} \, dx + \int \frac{1}{\sqrt{e^x}} \, dx \).

Step 2: Solve the First Integral

Focus on solving the first integral \( \int \frac{1}{\sqrt{x}} \, dx \). This integral is equivalent to \( \int x^{-1/2} \, dx \). Using the power rule for integration, \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( n = -\frac{1}{2} \), we get \( \frac{x^{1/2}}{1/2} + C = 2x^{1/2} + C \). Thus, the result is \( 2\sqrt{x} + C_1 \).

Step 3: Solve the Second Integral

Now solve the second integral \( \int \frac{1}{\sqrt{e^x}} \, dx \). This can be rewritten as \( \int e^{-x/2} \, dx \). Recognizing this as the integral of an exponential function, we use the formula \( \int e^{ax} \, dx = \frac{1}{a}e^{ax} + C \). Here, \( a = -\frac{1}{2} \), so the solution is \( \frac{1}{-1/2} e^{-x/2} + C = -2e^{-x/2} + C_2 \).

Step 4: Combine the Results

Add the results from Step 2 and Step 3, as the original problem was a sum of two integrals. Combine \( 2\sqrt{x} + C_1 \) and \( -2e^{-x/2} + C_2 \) to get: \( 2\sqrt{x} - 2e^{-x/2} + C \), where \( C \) is a constant representing the sum of \( C_1 \) and \( C_2 \).

Key Concepts

Power Rule for Integration

The power rule for integration is a fundamental technique in calculus. If you have a function of the form \( x^n \), you can easily find its indefinite integral using this rule. The formula for the power rule is:

• \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \)

where \( n eq -1 \) and \( C \) is the constant of integration. This rule is particularly handy for polynomials and functions that can be expressed in terms of power. In the exercise mentioned, for the integral \( \int \frac{1}{\sqrt{x}} \, dx \), we utilize the power rule by rewriting \( \frac{1}{\sqrt{x}} \) as \( x^{-1/2} \). Applying the power rule gives us the result \( 2\sqrt{x} + C_1 \). Breaking down exponents in this way not only simplifies the integration process but also strengthens your understanding of algebraic manipulations.

Exponential Functions

Exponential functions are a class of functions that involve the constant \( e \), the base of the natural logarithm, and exponents. These functions are unique because they have a constant relative rate of change; they grow or decay at a rate proportional to their value at any point. The general form of an exponential function is:

• \( f(x) = e^{ax} \)

where \( a \) is a constant. In integral calculus, integrating exponential functions is direct when using the formula:

• \( \int e^{ax} \, dx = \frac{1}{a}e^{ax} + C \)

For the integral \( \int \frac{1}{\sqrt{e^x}} \, dx \) from the exercise, we transformed it into \( \int e^{-x/2} \, dx \). By recognizing the form of an exponential function, we applied the integration formula and obtained \( -2e^{-x/2} + C_2 \). Mastering this technique allows you to handle a wide range of problems involving exponential decay and growth.

Integration Techniques

Integration is about finding a function whose derivative equals a given function. To do this effectively, one must be familiar with various integration techniques. These methods transform complex integrals into simpler ones that are more manageable. Some basic techniques include:

• Substitution: Used when an integral contains a function and its derivative. You replace a part of the integral to simplify it.
• Integration by Parts: Useful when integrating products of functions. It's based on the product rule for differentiation.

In the exercise, rather than applying these advanced techniques, we utilized basic transformations such as rewriting expressions to fit the power rule and recognizing exponential forms. These simpler approaches often suffice for straightforward integrals. Understanding when and how to use different techniques is crucial. It ensures that you select the most efficient method for a given problem, reducing errors and saving time.