Evaluate the integrals in Exercises 1-24 using integration by parts. ∫e^-y cosy d y

Name: Problem 22, Chapter 8: Calculus
Uploaded: 2021-01-20T02:32:10-08:00
Duration: 4 min 2 s
Channel: Madi Sousa
Description: Evaluate the integrals in Exercises 1-24 using integration by parts. ∫e^-y cosy d y

step 1 In this problem, we are practicing a new integration technique. Maybe you're learning this in cl

Evaluate the integrals in Exercises 1-24 using integration...

Key Concepts

Integration by Parts

Integration by parts is a method for integrating products of functions based on the product rule for differentiation. It involves choosing one function to differentiate (u) and another to integrate (dv), using the formula ? u dv = uv - ? v du. This technique simplifies the evaluation of integrals that are otherwise difficult to handle directly.

Recurrence Relations in Integration

When applying integration by parts repeatedly, the original integral may reoccur within the resulting expression. This creates a recurrence relation that can be solved algebraically by isolating the integral on one side of the equation, making it possible to determine the value of the integral.

Exponential Functions

Exponential functions, such as those involving e raised to a variable power, are unique because their derivatives and integrals are proportional to the function itself. This property makes them particularly amenable to integration by parts, although they may also contribute to recurrent terms when paired with other functions.

Trigonometric Functions

Trigonometric functions like cosine are characterized by their periodic nature and predictable derivative behavior. When integrated by parts, these functions often cycle through their sine and cosine forms, which can lead to a recurrence of the original integral, necessitating algebraic manipulation to resolve the integral.

Transcript

00:01 In this problem, we are practicing a new integration technique.

00:05 Maybe you're learning this in class and wants some extra practice, or maybe you're learning this for the first time and need to be walked through it.

00:14 And this problem focuses on what we call integration by parts.

00:18 It is a very useful technique for integration, and you're going to see it quite often.

00:24 So that's what we're going to be talking about.

00:25 So we are given the integral of e to the negative y times, a cosine of y in d .y.

00:33 Now looking at this list looks like a pretty hard integral to solve if i didn't know that i had any techniques on how to solve it.

00:42 So we're going to use integration by parts.

00:45 You can think of this almost like the product rule but for intervals.

00:50 That's kind of what we're getting at here.

00:53 So we have the integral of udv and that's going to be equal to uv minus the integral of v -d -u.

01:01 And we're going to pick what u and v are.

01:04 We are going to let u be e to the negative y.

01:09 Why is that? we can easily manipulate this value in order to cancel some terms or rearrange them very easily.

01:18 So then if we took the derivative, we would see that du would be negative e to the negative y and dy.

01:25 And now we have to pick something for v.

01:28 Now you might be asking, what do i choose for dv? essentially, that's everything that's left, and that would be the cosine of y and d y.

01:36 So then we need v by itself, so we would have to integrate this, and we'll find that v is the sign of y.

01:45 So now we can start plugging things in to this general formula...

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