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Chapter 4: Problem 17

Evaluate. Assume \(u>0\) when In \(u\) appears. (Be sure to check by differentiating!) $$\int t e^{-t^{2}} d t$$

Short Answer

Expert verified
-\frac{1}{2} e^{-t^{2}} + C

Step by step solution

01

- Set up the integral

Start with the given integral: \[ \int t e^{-t^{2}} d t \]
02

- Use substitution

Let鈥檚 use a substitution to simplify the integral. Set \( u = -t^{2} \). Therefore, \( d u = -2t d t \) which gives us \( -\frac{1}{2} du = t dt \).
03

- Substitute and simplify

Substitute \( u \) and \( du \) into the integral: \[ \int t e^{-t^{2}} d t = \int e^{u} \left(-\frac{1}{2}\right) d u \] This simplifies to: \[ -\frac{1}{2} \int e^{u} d u \]
04

- Integrate

Now, integrate \( e^{u} \): \[ -\frac{1}{2} \int e^{u} d u = -\frac{1}{2} e^{u} + C \]
05

- Substitute back

Recall that \( u = -t^{2} \). Substitute back: \[ -\frac{1}{2} e^{-t^{2}} + C \]
06

- Differentiate to check

Differentiate the result to confirm it matches the original integrand: \[ \frac{d}{d t} \left( -\frac{1}{2} e^{-t^{2}} \right) = e^{-t^{2}} (t) \] The differentiation checks out, confirming the solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Definite Integrals
Definite integrals represent the area under a curve over a specific interval. They are used to find quantities like areas, volumes, and total changes. For a function, the definite integral from a to b is written as \(\int_a^b f(x) \, dx\). This tells us to integrate the function over the limits a to b. After integration, we apply these limits to find the specific value. This is different from an indefinite integral, as it yields a numerical result rather than a function with a constant of integration.
Substitution Method
The Substitution Method is a powerful technique for evaluating integrals. It simplifies complex integrals by making a substitution. In this problem, we set \(\ u = -t^2\). Then, we compute the differential du. Since \(\ du = -2t \, dt\), we rewrite it as \(\ -\frac{1}{2} \ du = t \, dt\). By substituting these into the integral, we replace complex terms with simpler ones. This method often turns a complicated integral into a basic one we can easily solve.
Differentiation
Differentiation is the process of finding the derivative of a function. It's the reverse operation of integration. In our exercise, after finding the integral \(\ -\frac{1}{2} e^{-t^2} + C\), it's essential to differentiate to verify our solution. Calculating the derivative gives us the original function \(\ te^{-t^2}\), ensuring our integral solution is correct. Thus, differentiation helps confirm that the steps we took were accurate.
Indefinite Integrals
Indefinite integrals represent the collection of all antiderivatives of a function. Unlike definite integrals, they do not have limits. The result includes a constant of integration, represented as \(\ + C\), which accounts for all potential shifts of the antiderivative. In this problem, evaluating \(\ int t e^{-t^2} dt\) gives us \(\ -\frac{1}{2} e^{-t^2} + C\). This general form allows us to work with various initial conditions.

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