Problem 95 - Calculus and its applications

Question

Evaluate. $$ \int \frac{e^{\sqrt{t}}}{\sqrt{t}} d t $$

Step-by-Step Solution

Answer

The integral evaluates to \( 2e^{\sqrt{t}} + C \).

Step 1: Identify the substitution

You need to choose a substitution that will simplify the integral. Notice the term \( \sqrt{t} \) in both the numerator and the denominator. Use the substitution \( u = \sqrt{t} \), which implies \( t = u^2 \).

Step 2: Differentiate the substitution

Differentiate the substitution \( u = \sqrt{t} \) with respect to \( t \). This gives \( \frac{du}{dt} = \frac{1}{2\sqrt{t}} \), which rearranges to \( dt = 2u\,du \).

Step 3: Substitute into the integral

Replace \( \sqrt{t} \) and \( dt \) in the original integral with \( u \) and \( 2u\,du \) respectively. The integral becomes \( \int 2u \frac{e^u}{u} \ du = 2 \int e^u \ du \).

Step 4: Evaluate the new integral

The integral \( 2 \int e^u \ du \) is straightforward. The result is \( 2e^u + C \), where \( C \) is the integration constant.

Step 5: Substitute back to original variable

Replace \( u \) back with \( \sqrt{t} \) in the result from the previous step. This gives the final answer: \( 2e^{\sqrt{t}} + C \).

Key Concepts

Substitution Method

In integral calculus, the substitution method is a powerful tool that simplifies complex integrals by changing variables. This method involves replacing a part of the integral with a new variable, often making the process of integration more manageable. In our exercise, we identified that using substitution could simplify our integral \( \int \frac{e^{\sqrt{t}}}{\sqrt{t}} \, dt \).
Here's how it works in practice:

• First, identify a substitution that will simplify the integral. Here, we chose \( u = \sqrt{t} \). This means wherever we see \( \sqrt{t} \), we replace it with \( u \).
• Next, determine the differential substitution by differentiating \( u = \sqrt{t} \). This gives us \( dt = 2u \, du \), which can replace \( dt \) in the integral.

By making these substitutions, the complex expression becomes simpler and easier to integrate. The goal is to transform the integral into a standard form where direct integration methods are applicable.

Indefinite Integrals

Indefinite integrals, often referred to as antiderivatives, represent a family of functions whose derivative is the function being integrated. An indefinite integral is expressed with the integral sign \( \int \), followed by the function and the differential of the variable.
The original exercise involves evaluating an indefinite integral. Our integral function is \( \frac{e^{\sqrt{t}}}{\sqrt{t}} \), and through substitution, it transforms into a simple expression \( 2 \int e^u \, du \).
Important concepts to remember:

• The integral sign \( \int \) indicates the operation of integration, while the constant \( C \) represents any constant that could be added to the antiderivative, accounting for all possible functions that differentiate to the given function.
• In our solution, after integration, \( C \) is added to give \( 2e^{u} + C \).

Indefinite integrals are essential for understanding the broader picture of the area under the curve and cumulative quantities.

Integration Techniques

Integration in calculus is about finding the antiderivative of a function or determining the area under a curve. Various techniques can be employed depending on the function at hand, and each technique simplifies the integration process under different circumstances.
Among the most used techniques are:

• Substitution: It's like reverse chain rule; useful when there's a composite function in the integrand.
• Integration by Parts: Handy for products of functions.
• Partial Fractions: Useful for rational functions with polynomials.

In our exercise, substitution was the most suitable technique as it simplified the integral into a more manageable form. The transformed integrand, \( e^u \), became trivial to integrate using basic rules of calculus, resulting in \( 2e^{u} \). After finding the antiderivative, we returned to the original variable to express the solution, highlighting the effectiveness and applicability of different integration techniques.