Question

Evaluate the definite integral. $$ \int_{1}^{3} \frac{e^{3 / x}}{x^{2}} d x $$

Short answer

The value of the definite integral is \(e^3 - e^1\).

Step-by-step solution

Identify the substitution

Look for a function and its derivative in the integrand. Here you can spot that the derivative of 3/x is -3/x^2 up to a constant. Therefore, let the substitution variable be u=3/x. Find du/dx.= -3/x^2, and rewrite it as dx = -du/(3u^2)

Apply the substitution

Substitute everything in terms of 'u' into the integrand. Also, don't forget to change the limits of integration from x to u using the relation u=3/x. For x=1, u=3 and for x=3, u=1. The definite integral becomes \(-\int_{3}^{1} e^{u} du\).

Compute the antiderivative

Compute the antiderivative of e^u which is also e^u. So, we have \(-\left[e^{u}\right]_{3}^{1}\)

Evaluate the limits of the antiderivative

Evaluate the antiderivative at the upper limit and at the lower limit then subtract them: -[e^1 - e^3] = e^3 - e^1

Key concepts

Substitution Method

The substitution method is a powerful tool used in calculus to simplify the process of finding integrals, especially when dealing with complex expressions. By using substitution, you essentially change the variable of integration, turning the integrand into an easier form to work with.
To begin, you find a part of the integrand where a substitution can be made. Look for a function and its derivative within the integral. In our exercise, we identified the substitution as \( u = \frac{3}{x} \). Detecting this relationship helps break down the problem into simpler parts.
Once the substitution is chosen, calculate \( \frac{du}{dx} \) and solve for \( dx \) in terms of \( du \). This helps in replacing all the components in the original integral. Don't forget to adjust the limits of integration to correspond with your new variable \( u \).

• Identify substitution: Find a function and its potential derivative.
• Change variables: Connect your substitution choice and adjust integral limits.

Antiderivative

The antiderivative is a fundamental concept in calculus, closely related to the process of integration. It is essentially the reverse of differentiation. Finding an antiderivative allows us to evaluate the original function over an interval if it exists.
In our exercise, after making a substitution, we are left with the straightforward problem of finding the antiderivative of \( e^u \). The antiderivative of \( e^u \) is \( e^u + C \), where \( C \) is the constant of integration. However, since we are dealing with a definite integral, the constant \( C \) isn't needed as it cancels out.
Knowing the antiderivative of \( e^u \) allows us to find a simple form that can be evaluated over the specified limits. This part is crucial because it's where we truly convert the substitution effort into a solution.

• Retrieve original function: Reverse the differentiation process.
• Apply substitution: Use information from substitution to work with the antiderivative.

Limits of Integration

The limits of integration play a key role when evaluating a definite integral. Initially, they define the range over which you want to integrate your function. In the substitution method, these limits need careful translation to the new variable.
In our problem, once the substitution \( u = \frac{3}{x} \) is made, the limits of integration, originally from \( x = 1 \) to \( x = 3 \), should change. Calculate the new limits by substituting the original limits into the substitution formula:

• For \( x = 1 \), \( u = \frac{3}{1} = 3 \)
• For \( x = 3 \), \( u = \frac{3}{3} = 1 \)

Now the limits of integration turn from \( 3 \) to \( 1 \). This change is crucial and must be handled meticulously to ensure the correct evaluation.
Finally, evaluate the antiderivative at the new limits: substitute these limits into the antiderivative and subtract the lower limit evaluation from the upper limit. In our example, this process gives the final result of \( e^3 - e^1 \).

• Adjust limits for the substitution: Calculate the new bounds.
• Ensure proper evaluation: Use the new limits for calculations.