Problem 45 - Calculus for Biology and Medicine

Question

First make an appropriate substitution and then use integration by parts to evaluate the indefinite integrals. $$ \int_{0}^{1} e^{\sqrt{x}} d x $$

Step-by-Step Solution

Answer

The integral evaluates to 2.

Step 1: Identify the substitution

To simplify the integral \( \int_{0}^{1} e^{\sqrt{x}} dx \), we can introduce a substitution. Set \( u = \sqrt{x} \). Therefore, \( x = u^2 \) and the differential \( dx = 2u \, du \).

Step 2: Change the limits of integration

With the substitution \( u = \sqrt{x} \), when \( x = 0 \), \( u = 0 \) and when \( x = 1 \), \( u = 1 \). The integral now becomes \( \int_{0}^{1} e^u \cdot 2u \, du \).

Step 3: Set up for integration by parts

Use integration by parts, where \( \int u \, dv = uv - \int v \, du \). Choose \( u = 2u \) and \( dv = e^u \, du \). Then, differentiate \( 2u \) to get \( du = 2 \, du \), and integrate \( e^u \) to get \( v = e^u \).

Step 4: Apply integration by parts formula

Substitute into the integration by parts formula: \( \int 2u \, e^u \, du = 2u \cdot e^u \bigg|_{0}^{1} - \int e^u \cdot 2 \, du \).

Step 5: Simplify and integrate remaining integral

Evaluate the boundary term: \( 2u \cdot e^u \bigg|_{0}^{1} = [2 \cdot 1 \cdot e^1 - 2 \cdot 0 \cdot e^0] = 2e \). The remaining integral is \( -2 \int e^u \, du = -2e^u \bigg|_{0}^{1} = -2(e - 1) \).

Step 6: Combine results

Combine the evaluated parts: \( 2e - 2(e - 1) = 2e - 2e + 2 = 2 \).

Step 7: Conclude solution

The value of the integral \( \int_{0}^{1} e^{\sqrt{x}} dx \) is \( 2 \).

Key Concepts

Integration by Parts

Integration by parts is a powerful technique to solve integrals, particularly when faced with the product of two functions. The formula for integration by parts is derived from the product rule for differentiation and is represented by:\[\int u \, dv = uv - \int v \, du\]To effectively use this strategy, you select which part of the integral's components will be \( u \) and which will be \( dv \). A general suggestion is to choose \( u \) such that its derivative \( du \) is simpler than \( u \), and \( dv \) such that it can be easily integrated to find \( v \).

• Choose \( u \) to be a function that becomes simpler when differentiated.
• Choose \( dv \) to be easily integrable.

In the exercise, after the substitution, the integral \( \int 2u \cdot e^u \, du \) required selecting parts: \( u = 2u \) and \( dv = e^u \, du \). This strategic choice simplifies the process by allowing the \( e^u \) to remain unchanged, given its straightforward nature. Once functions are chosen correctly, remember to use the formula step-by-step as demonstrated, reducing the overall complexity of the integral.

Indefinite Integrals

Indefinite integrals represent a family of functions and are crucial for finding antiderivatives. When you integrate a function without bounds, the result is an indefinite integral, symbolically written without specific limits of integration. It is the reverse process of differentiation.The key aim here is to determine the antiderivative of the function provided. For indefinite integrals, always remember to add the constant of integration \( C \), reflecting the family of possible functions. This constant is essential because differentiation removes it, meaning all functions differing by a constant have the same derivative.

• An indefinite integral's solution always includes \( C \).
• The scope is finding generalized functions without specific limits.

In the current problem's context, even though the procedure ends with a specific numerical value due to evaluation bounds, the steps leading to integration by parts first deal with finding the antiderivative.

Substitution Method

The substitution method is a key technique in calculus used to simplify complex integrals by changing variables. This method is similar to the chain rule used in differentiation. It is particularly useful when an integral contains a composition of functions.To use the substitution method:1. Identify a substitution that simplifies the integral. This usually involves setting \( u \) equal to some expression inside the integral. The goal is to reduce complexity by turning a composite function into a basic one.2. Compute the differential of \( u \) to replace \( dx \), turning the original integral into one in terms of \( u \).3. Change the limits of integration if the integral is definite.In the problem given, the substitution \( u = \sqrt{x} \) converts \( dx \) into \( 2u \, du \), simplifying the complex original integral \( \int e^{\sqrt{x}} \, dx \) into a more manageable form \( \int e^u \cdot 2u \, du \). By successfully applying substitution, the problem becomes more straightforward, often transforming it into a standard form solvable by other techniques like integration by parts.