Problem 53 - Applied Calculus

Question

Use integration by substitution and the Fundamental Theorem to evaluate the definite integrals. $$\int_{1}^{4} \frac{e^{\sqrt{x}}}{\sqrt{x}} d x$$

Step-by-Step Solution

Answer

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

Step 1: Identify the substitution

Recognize that the integrand contains terms of the form \( e^{ ext{(something)}} \) and \( rac{1}{ ext{(same something)}} \). Set \( u = \sqrt{x} \), which implies that \( x = u^2 \). This implies \( dx = 2u \, du \).

Step 2: Change limits of integration

With the substitution \( u = \sqrt{x} \), change the limits of integration from \( x \) values to \( u \) values. When \( x = 1 \), \( u = \sqrt{1} = 1 \). When \( x = 4 \), \( u = \sqrt{4} = 2 \). So, the limits change from 1 to 4, to 1 to 2.

Step 3: Substitute and rewrite the integral

Substitute the expressions for \( u \), \( dx \), and change the limits, rewriting the integral:\[ \int_{1}^{4} \frac{e^{\sqrt{x}}}{\sqrt{x}} \, dx = \int_{1}^{2} e^u \, \frac{1}{u} (2u \, du) = 2 \int_{1}^{2} e^u \, du \].

Step 4: Integrate with respect to u

Integrate \( 2 \int e^u \, du \). The antiderivative of \( e^u \) is \( e^u \). Hence:\[ 2 \int e^u \ du = 2e^u \], evaluated from \( u=1 \) to \( u=2 \).

Step 5: Apply the Fundamental Theorem of Calculus

Use the Fundamental Theorem of Calculus to evaluate the integral after finding the antiderivative:\[ 2[e^u]_{1}^{2} = 2[e^2 - e^1] = 2(e^2 - e) \].

Step 6: Simplify the result

The result of the definite integral, after simplification, is:\[ 2(e^2 - e) = 2e^2 - 2e \].

Key Concepts

Definite Integrals

Definite integrals are a fundamental concept in calculus, representing the exact area under a curve over a specified interval. In our exercise, we're dealing with the integral of the function \( \frac{e^{\sqrt{x}}}{\sqrt{x}} \) from 1 to 4.
This means we're calculating the area under the curve defined by this function between \( x = 1 \) and \( x = 4 \).
The notation for a definite integral, \( \int_{a}^{b} f(x) \, dx \), includes two key components: the interval \([a, b]\) over which we integrate, and the function \( f(x) \) itself.

• The lower limit, \( a \), represents where the integration begins.
• The upper limit, \( b \), is where it ends.

Definite integrals can be evaluated directly or through various techniques, such as substitution, which helps simplify the arithmetic or algebra involved.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus bridges the concepts of differentiation and integration. It states that the process of differentiation and integration are, in a sense, inverse operations.
This theorem has two parts:

• First, it guarantees that every continuous function has an antiderivative. This means you can "reverse" differentiation to obtain a function whose derivative is the given function.
• Second, it provides a way to compute a definite integral using antiderivatives.

In our problem, once we've found the antiderivative of \( e^u \), we use the theorem to evaluate it at the limits \( u = 1 \) and \( u = 2 \).
We subtract the value of the function at the lower limit from the value at the upper limit, effectively finding the net "change" represented by our integral.

Change of Variables

The change of variables is a technique used to simplify integrals, especially when the function inside the integral is complex or cumbersome.
In this exercise, we used a substitution method where we let \( u = \sqrt{x} \), which transformed the variable of integration from \( x \) to \( u \).

• This substitution required us to also change the differential \( dx \) and the limits of integration to match our new variable \( u \).
• \( dx = 2u \, du \) is obtained from differentiating \( u = \sqrt{x} \).
• Our limits changed from \( x \, \text{values} \) to \( u \, \text{values} \): from 1 to 4 in \( x \) to 1 to 2 in \( u \).

This simplification allows us to handle the integral in a more straightforward manner.

Antiderivatives

Antiderivatives are functions that "undo" the process of differentiation. When you're given a function \( f(x) \), its antiderivative is a function \( F(x) \) such that \( F'(x) = f(x) \).
For the exercise's integral, our challenge was to find the antiderivative of the function \( e^u \), which is simply \( e^u \).
Once we have this antiderivative, we apply it to evaluate our definite integral.

• The process involves substituting the limits into the antiderivative \( 2[e^u]_{1}^{2} \).
• The Fundamental Theorem of Calculus then lets us find the difference \( e^2 - e \).
• Finally, multiplying by the constant factor of 2 gives the final result of \( 2(e^2 - e) \).

This key concept of finding antiderivatives is integral to solving many problems in calculus seamlessly.