Problem 25 - College Algebra and Calculus: An Applied Approach

Question

Find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.) $$ \int \frac{e^{1 / t}}{t^{2}} d t $$

Step-by-Step Solution

Answer

The indefinite integral of \(\frac{e^{1 / t}}{t^{2}}\) with respect to \(t\) is \(-e^{1/t} + C\).

Step 1: Identify the Proper Substitution

For \(\int e^{1/t}/t^{2} dt\), notice the derivative of \(1/t\), which is \(-1/t^{2}\), is in the integrand. That signifies that we should set \(u = 1/t\), therefore, \(du = -1/t^{2} dt\). Change of variables gives us a simpler integral.

Step 2: Substitute the Variables

Substitute \(u = 1/t\) and \(du = -1/t^{2} dt\), the integral becomes \(-∫e^{u} du\), which looks much simpler.

Step 3: Compute the Indefinite Integral

An antiderivative of the function \(e^{u}\) is \(e^{u}\), hence the integral of \(-e^{u}\) is \(-e^{u}\) + C, where C is the constant of integration.

Step 4: Reverse Substitution

Substitute \(u = 1/t\) back in to obtain the final answer: \(-e^{1/t} + C\).

Key Concepts

Substitution Method

The substitution method in integration is a powerful technique that simplifies complex integrals. It involves changing the variables in an integral to make the integration process easier.
When confronted with a difficult integral, identifying a part of the integrand that can be substituted is crucial. In the given problem, we notice that the derivative of \(1/t\), which is \(-1/t^2\), happens to be a part of the integrand \(e^{1/t}/t^2\). This suggests using a substitution where \(u = 1/t\).

• Set \(u = 1/t\) and find \(du\). Here, \(du = -1/t^2 dt\).
• Replace \(dt\) and transform the integral into an easier form by substituting the variables.

The substitution simplifies the original integral to \(-\int e^{u} du\). This new integral is easier to solve.

Antiderivative

The antiderivative, or the indefinite integral, of a function is a fundamental part of calculus. It represents a reverse operation to differentiation.

Finding an antiderivative means identifying a function whose derivative gives the original function. In the context of the problem, once we've substituted and simplified our integral, we are tasked with finding the antiderivative of \(e^u\).

• The antiderivative of \(e^u\) is \(e^u\), because the derivative of \(e^u\) is itself \(e^u\).
• Don't forget to add the constant of integration \(C\) when finding indefinite integrals. This constant accounts for any vertical shifts in the family of functions.

Thus, the integral becomes \(-e^u + C\) after calculating the antiderivative and adding the constant.

Integration by Parts

Integration by parts is another technique in calculus used to evaluate integrals, particularly when products of functions are involved.
However, in the context of this particular problem, integration by parts was not necessary due to the choice of substitution which significantly simplified the integral.

Nonetheless, understanding integration by parts can be useful in other scenarios. It is based on the principle of the product rule for differentiation. The formula can be stated as follows:
\[\int u \, dv = uv - \int v \, du\]Here are the steps for using this method:

• Choose \(u\) and \(dv\) from the integral \(\int u \, dv\).
• Differentiate \(u\) to get \(du\) and integrate \(dv\) to get \(v\).
• Substitute into the formula to simplify and solve the integral.

In our solved problem, the substitution method already simplified the integral enough, making integration by parts unnecessary for this specific case.