Problem 95 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

\( \frac{2}{3} x^{3/2} + 2 e^{x/2} + C \) is the indefinite integral.

Step 1: Break Down the Integral

Separate the given integral into two distinct integrals: \( \int \sqrt{x} \, dx + \int \sqrt{e^x} \, dx \). This allows us to address each part individually.

Step 2: Integrate \( \sqrt{x} \)

The function \( \sqrt{x} \) can be rewritten as \( x^{1/2} \). The power rule for integration states that \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \). Applying this rule:\[\int x^{1/2} \, dx = \frac{x^{3/2}}{3/2} + C = \frac{2}{3} x^{3/2} + C\]

Step 3: Integrate \( \sqrt{e^x} \)

In this part, \( \sqrt{e^x} = e^{x/2} \). Again, using the power rule and considering the derivative of the exponent:\[\int e^{x/2} \, dx = \int e^{x/2} \cdot 2 \, d(x/2) = 2 \cdot e^{x/2} + C\]The factor of 2 accounts for the chain rule.

Step 4: Combine the Results

Combine the integrals of the two separate functions:\[\int \sqrt{x} \, dx + \int \sqrt{e^x} \, dx = \frac{2}{3} x^{3/2} + 2 e^{x/2} + C\] where \( C \) is the constant of integration.

Key Concepts

Indefinite Integrals

Indefinite integrals are a fundamental concept in integral calculus. Unlike definite integrals, which calculate a specific area between a function and the x-axis over a defined range, indefinite integrals represent a family of functions. Indefinite integrals include an arbitrary constant, denoted as \( C \), because integration is an inverse operation of differentiation and we lose information about the constant.
In mathematical terms, if \( F(x) \) is the antiderivative of \( f(x) \), then the indefinite integral of \( f(x) \) is written as:

• \( \int f(x) \, dx = F(x) + C \)

The goal is to find this antiderivative, which can describe the accumulation rate of \( f(x) \). In our exercise, we compute the indefinite integrals of \( \sqrt{x} \) and \( \sqrt{e^x} \), each developing into its own family of functions after integration. These results are combined with a constant to represent the general solution.

Power Rule

The power rule is a powerful tool in calculus that simplifies the process of finding antiderivatives or indefinite integrals of polynomials and functions that can be rewritten in a polynomial form.
This rule states:

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

This formula is applicable wherever \( x \) has a constant exponent \( n \). In the context of our exercise, we use the power rule to determine the integral of \( x^{1/2} \) by flipping the exponent:

• \( \int x^{1/2} \, dx = \frac{x^{1/2+1}}{1/2+1} = \frac{2}{3} x^{3/2} + C \)

This manipulation allows easy transition from function to its antiderivative form. The power rule can also be adapted to integrate exponential functions by using algebraic manipulation before applying it.

Integration by Substitution

Integration by substitution, often compared to the chain rule in differentiation, simplifies complex integrals by changing the variable of integration. This technique transforms the original integral to a simpler form where the power rule or basic methods can be easily applied.
In our exercise, for the integral of \( \sqrt{e^x} \), we first rearrange the function to \( e^{x/2} \). Using substitution:

• Let \( u = x/2 \), then \( du = 1/2 \, dx \), or \( dx = 2 \, du \)

The integral becomes simpler:

• \( \int e^{x/2} \, dx = \int e^u \cdot 2 \, du = 2 \int e^u \, du \)

Integrating \( e^u \) simplifies to \( e^u \), leading us back to the original variable with:

• \( 2e^{x/2} + C \)

This method is particularly useful for more challenging integrals and highlights the importance of manipulation before integration.