Problem 51 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

\( \frac{2}{3} x^{3/2} + 2x^{1/2} + C \)

Step 1: Rewrite the Expression

Begin by rewriting the integral \[ \int \left( \sqrt{x} + \frac{1}{\sqrt{x}} \right) \, dx \] in terms of exponents. Recall that \(\sqrt{x} = x^{1/2}\) and \(\frac{1}{\sqrt{x}} = x^{-1/2}\). This transforms the integral into:\[ \int \left( x^{1/2} + x^{-1/2} \right) \, dx \]

Step 2: Separate the Integral

Break down the integral into two separate integrals using the linearity of integration:\[ \int x^{1/2} \, dx + \int x^{-1/2} \, dx \]

Step 3: Apply the Power Rule to Each Integral

Use the power rule for integrals, which states \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\), where \(C\) is the constant of integration.- For \(\int x^{1/2} \, dx\): The new exponent is \(1/2 + 1 = 3/2\). So, \(\int x^{1/2} \, dx = \frac{x^{3/2}}{3/2} = \frac{2}{3} x^{3/2}\).- For \(\int x^{-1/2} \, dx\): The new exponent is \(-1/2 + 1 = 1/2\). So, \(\int x^{-1/2} \, dx = \frac{x^{1/2}}{1/2} = 2x^{1/2}\).

Step 4: Combine the Results and Include the Constant of Integration

Combine the solutions from each integral and include the constant of integration:\[ \frac{2}{3} x^{3/2} + 2x^{1/2} + C \]Thus, the indefinite integral is \[ \int \left( \sqrt{x} + \frac{1}{\sqrt{x}} \right) \, dx = \frac{2}{3} x^{3/2} + 2x^{1/2} + C \]

Key Concepts

Power Rule for Integrals

The power rule for integrals is a fundamental concept that makes integrating expressions with exponents straightforward. When you see an integral in the form \( \int x^n \, dx \), the power rule guides you on how to integrate it by increasing the exponent by one and then dividing by the new exponent.

Here's the formula: - \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \). The letter \( C \) here represents the constant of integration, which we will discuss in another section. This formula works for any real number \( n \) except \( n = -1 \), as that case requires a different rule involving logarithms.

In our exercise, the expression \( \sqrt{x} \) is rewritten as \( x^{1/2} \), and \( \frac{1}{\sqrt{x}} \) is written as \( x^{-1/2} \). Integrating these using the power rule:

• For \( \int x^{1/2} \, dx \), increase the exponent by one: \( 1/2 + 1 = 3/2 \), and divide by the new exponent, giving you \( \frac{x^{3/2}}{3/2} \), which can be simplified to \( \frac{2}{3} x^{3/2} \).
• For \( \int x^{-1/2} \, dx \), increase the exponent by one: \( -1/2 + 1 = 1/2 \), and divide by the new exponent, resulting in \( \frac{x^{1/2}}{1/2} \), simplifying to \( 2x^{1/2} \).

Linearity of Integration

The linearity of integration property is a powerful tool that makes handling integrals simpler. This rule states that the integral of a sum is the sum of the integrals; it allows us to integrate each term separately.

In mathematical terms, if you have an integral of a sum like \( \int (f(x) + g(x)) \, dx \), this can be split into:- \( \int f(x) \, dx + \int g(x) \, dx \).
For the given problem, \( \int (\sqrt{x} + \frac{1}{\sqrt{x}}) \, dx \), the expression can be broken down using this principle.

So, it becomes:

• \( \int \sqrt{x} \, dx + \int \frac{1}{\sqrt{x}} \, dx \)

Using linearity allows us to focus on one term at a time, making the integral much easier to solve. After breaking them down, each part can be solved using the power rule for integrals, as outlined in the previous section.

Constant of Integration

When we deal with indefinite integrals, you'll always encounter a term called the constant of integration, symbolized by \( C \). This constant is crucial because it accounts for the family of functions that share the same derivative.

This means that when you take the derivative of any function in this family, you get the original function inside the integral. When computing the indefinite integral \( \int x^n \, dx \), adding \( C \) is necessary because the process of integration "forgets" what constant might have been there before differentiation.

In our exercise, after separately integrating \( \int x^{1/2} \, dx \) and \( \int x^{-1/2} \, dx \), we combine their results and include \( C \) at the end:

• Resulting in \( \frac{2}{3} x^{3/2} + 2x^{1/2} + C \)

Remember, \( C \) can be any real number, representing the constant part of the family of all possible antiderivatives. Always include \( C \) in indefinite integrals to ensure that the solution accommodates all potential initial conditions.