Problem 39 - Thomas Calculus in SI Units

Question

Evaluate the integrals in Exercises \(29-50.\) $$\int 2 t e^{-t^{2}} d t$$

Step-by-Step Solution

Answer

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

Step 1: Identify the Type of Integral

This integral is of the type that can be solved using the substitution method. Specifically, the integrand is a product of a function and its derivative.

Step 2: Choose an Appropriate Substitution

Let us choose the substitution \( u = -t^2 \). Then the derivative is \( du = -2t \, dt \), which simplifies to \( -du = 2t \, dt \).

Step 3: Rewrite the Integral in Terms of the New Variable

Substitute \( u = -t^2 \) and \( 2t \, dt = -du \) into the integral. It becomes: \(-\int e^u \, du \).

Step 4: Integrate Using Basic Integration Rules

Since \(-\int e^u \, du\) is a standard integral, its integral is \(-e^u + C\), where \(C\) is the constant of integration.

Step 5: Substitute Back the Original Variable

Replace \(u\) with \(-t^2\) to get the final result: \(-e^{-t^2} + C\).

Key Concepts

Substitution Method

The substitution method is a popular technique in integral calculus that simplifies the process of integration. This method involves changing variables to make integration easier.
When you have an integral that includes a function and its derivative, substitution can be particularly effective.

• Start by identifying a part of the integrand that can be substituted with a single variable, let's say, "u."
• Find the derivative of this chosen part, and ensure that the rest of the integrand contains this derivative as a factor.

In the given exercise, we used the substitution \( u = -t^2 \), transforming the integral into one that's simpler to evaluate. By substituting correctly, we turn a complex integral into one involving exponential functions of the form \(-\int e^u \, du \).
This approach not only makes integration straightforward but also helps in building a clearer understanding of how functions relate to their derivatives in calculations.

Exponential Functions

Exponential functions are a key component in many areas of mathematics. In integral calculus, they often appear due to their unique property of being proportional to their derivatives.
The function \( e^x \) is particularly special because its derivative is itself. This characteristic plays an essential role in simplifying integration problems involving exponential functions.

• In the particular problem, the expression \( e^{-t^2} \) becomes \( e^u \) after substitution.
• While \(-t^2 \) changing to \( u \) maintains the structure of an exponential function.

Exponential functions consistently appear in real-world phenomena, such as growth models or radioactive decay, making them a valuable tool in both theoretical and applied mathematics. Understanding their role in integrals can make tackling various calculus problems more intuitive.

Integration Techniques

Integration techniques are diverse methods employed to solve definite or indefinite integrals. Integral calculus often requires selecting the right technique to simplify and solve an integral. Understanding a few key strategies can significantly enhance your ability to approach these problems:

• Substitution: As previously discussed, this technique is excellent for integrals involving a function and its derivative.
• Integration by Parts: Useful for products of functions that don’t fit easily with substitution.
• Trigonometric Substitution: Optimizes integrals involving radicals and trigonometric identities.
• Partial Fraction Decomposition: Effective for rational functions where the numerator’s degree is less than the denominator’s.

Choosing the right technique often comes with practice and familiarity with different types of integrands. In this exercise, substitution proved ideal for simplifying and solving the integral of an exponential function. Integrating effectively requires balancing the structural qualities of an integrand with the appropriate mathematical method.