University:
Whitman College
Course:
MATH-126 | Calculus II
Academic year:

2024
Views:

200

Pages:

2
Author:

Jade Rosario

Integration By Parts: Using a Table Often, we need to do integration by parts several times to obtain an antiderivative. This shows you how to do it using a table, and you will find it very convenient. Note, however, there are times when a table shouldn’t be used, and we’ll see examples of that as well. The general idea: Integration by parts uses the formula: Z Z u dv = uv − v du Notice that this can be put into a table: sign Differentiate Antidifferentiate + u dv du v where we get the same formula by going diagonally from u, multiply by v, and then take the minus sign and integrate the product going straight across. We’ll show how Z to use the chart by the use of several examples: Example 1: x2 e−2x dx We’ll use the chart, and differentiate on x2 : + x2 e−2x − 2x − 12 e−2x + 2 1 −2x 4e − 0 − 18 e−2x Example 2: + x Z 1 1 1 1 ⇒ − x2 e−2x − xe−2x − e−2x + 2 2 4 8 Z 0 · e−2x dx x sin(2x) dx sin(2x) − 1 1 1 1 − 12 cos(2x) ⇒ − x cos(2x) + sin(2x) − 2 4 4 + 0 − 14 sin(2x) 1 Z 0 · sin(2x) dx Example 3: Z e2x sin(3x) dx This is a special case. We know we’ll have to integrate by parts twice, then look at what we get: + e2x − 2e 2x + 4e2x sin(3x) − 13 1 2 4 cos(3x) ⇒ − e2x cos(3x) + e2x sin(3x) − 3 9 9 Z e2x sin(3x) dx − 19 sin(3x) Rewriting this, we see that: Z Z 1 2x 2 2x 4 2x e sin(3x) dx = − e cos(3x) + e sin(3x) − e2x sin(3x) dx 3 9 9 so that: 13 9 Z 1 2 e2x sin(3x) dx = − e2x cos(3x) + e2x sin(3x) 3 9 And we arrive at the final answer: Z 3 2 e2x sin(3x) dx = − e2x cos(3x) + e2x sin(3x) 13 13 2

Integration By Parts Using a Table

Related Documents

Description

Related Documents

Free up your schedule!

Take 5 seconds to unlock