Problem 65 - Calculus for Biology and Medicine

Question

Evaluate each definite integral. $$ \int_{1}^{4} e^{\sqrt{x}} d x $$

Step-by-Step Solution

Answer

The integral evaluates to \(2(e^2 + e - 1)\).

Step 1: Recognize the Integral

The integral to evaluate is \( \int_{1}^{4} e^{\sqrt{x}} \, dx \). This is a definite integral from \(x = 1\) to \(x = 4\).

Step 2: Identify the Substitution

Notice that the integrand \( e^{\sqrt{x}} \) involves a function of \( x \) inside the exponential. A good technique is substitution. Let's set \( u = \sqrt{x} \), which implies \( x = u^2 \).

Step 3: Find the Derivative and Differential

Differentiate \( u = \sqrt{x} \) to find \( dx \) in terms of \( du \). The derivative \( \frac{du}{dx} = \frac{1}{2\sqrt{x}} = \frac{1}{2u} \). Therefore, \( dx = 2u \, du \).

Step 4: Change the Limits of Integration

When \( x = 1 \), \( u = \sqrt{1} = 1 \). When \( x = 4 \), \( u = \sqrt{4} = 2 \). So, the new limits for \( u \) are 1 to 2.

Step 5: Rewrite the Integral

Substitute \( u \) and \( dx \) in the integral: \( \int_{1}^{4} e^{\sqrt{x}} \, dx = \int_{1}^{2} e^{u} \cdot 2u \, du \). Simplify to \( 2 \int_{1}^{2} ue^{u} \, du \).

Step 6: Integration by Parts

To solve \( \int ue^{u} \ du \), use integration by parts: \( \int u \, dv = uv - \int v \, du \). Let \( u = u \) and \( dv = e^{u} \, du \), then \( du = du \) and \( v = e^{u} \).

Step 7: Apply Integration by Parts

Apply the formula: \[ \int ue^{u} \, du = ue^{u} - \int e^{u} \, du = ue^{u} - e^{u} + C \]

Step 8: Evaluate the Definite Integral

Now evaluate \( 2 \times ( ue^u - e^u ) \) from 1 to 2: \[ 2 \left[ (2e^2 - e^2) - (1e^1 - e^1) \right] \] Simplify inside the brackets to get \( 2 \left[ e^2 - (1 - e) \right] = 2 \left[ e^2 - 1 + e \right] \).

Step 9: Final Calculation

Compute the result: \( 2(e^2 + e - 1) \). This is the final value of the definite integral.

Key Concepts

Integration by Parts

Integration by parts is a powerful technique deriving from the product rule for differentiation. It's particularly useful when the integrand is a product of two functions. The formula for integration by parts is:\[ \int u \, dv = uv - \int v \, du \]In our example, after substituting variables, we arrive at the integrand \( ue^u \). Here, choose \( u = u \) and \( dv = e^u \, du \). This allows us to differentiate \( u \) to get \( du \), and integrate \( e^u \) to obtain \( v = e^u \).Applying the integration by parts formula results in:

• \( \int ue^u \, du = ue^u - \int e^u \, du = ue^u - e^u + C \)

This expresses the integration of a function that appears exponentially complex into simpler elements that can be straightforwardly computed. Integration by parts often breaks a difficult integral into manageable components. For definite integrals, as here, don't forget to apply calculated limits afterward.

Substitution Method

The substitution method, also known as \emph{u-substitution}, helps solve integrals by simplifying the integrand into a form that is easier to integrate. By replacing complex inner functions with a single variable, we can often turn intricate integrals into simpler, more standard forms.In the problem, the original integrand \( e^{\sqrt{x}} \) suggested a substitution. We set \( u = \sqrt{x} \), making it possible to replace \( x \) with \( u^2 \). Then, differentiate \( u = \sqrt{x} \) to find the corresponding differential:

• \( dx = 2u \, du \)

In this substitution:

• Original integral limits change. When \( x = 1 \), \( u = 1 \). When \( x = 4 \), \( u = 2 \).
• The integral \( \int_{1}^{4} e^{\sqrt{x}} \, dx \) transforms into \( 2 \int_{1}^{2} ue^{u} \, du \).

Substitution permits us to reframe the problem into a form that is now solvable using standard integration techniques, like integration by parts in this scenario.

Calculating Limits of Integration

When performing definite integrals, setting up the correct limits post-substitution is crucial. Computing limits correctly ensures the integral you've transformed stays consistent with its bounds.Once substitution is made with \( u = \sqrt{x} \), the need arises to convert the original bounds of integration to this new variable. Calculating these limits involves simply substituting old limits into the substitution expression:

• For initial \( x = 1 \), find \( u = \sqrt{1} = 1 \).
• For terminal \( x = 4 \), compute \( u = \sqrt{4} = 2 \).

Post substitution, your integral’s limits become \( u = 1 \) to \( u = 2 \).Finally, ensure these limits reflect the entire range of integration correctly after transformation. This accuracy enables the computation of correct and reliable solutions for the definite integral. Remember, changing of variables often implies changing limits accordingly, which is fundamental to integrating properly over the desired range.