Topic: Integration by Parts
Integration by Parts
We know how to evaluate
3
න𝑥 𝑑𝑥 ,
න sin 𝑥 𝑑𝑥,
න𝑥𝑒
𝑥2
𝑑𝑥
What about
න𝑥 2 𝑒 𝑥 𝑑𝑥 ?
Let’s write the product rule for derivatives and integrate.
𝑑
𝑓 𝑥 𝑔 𝑥 =
𝑑𝑥
Integration by Parts
න𝑢 𝑑𝑣 = 𝑢𝑣 − න𝑣𝑑𝑢
Choosing 𝑢 makes a difference. A mnemonic for remembering how to choose 𝑢
is ILATE
Example
Find
න7𝑥 cos 2𝑥 𝑑𝑥 .
Example
Find
න𝑥 2 𝑒 𝑥 𝑑𝑥 .
Example
Evaluate
න𝑥 5 sin 𝑥 𝑑𝑥 .
Example
Evaluate
නln 𝑥 𝑑𝑥 .
Example
Calculate.
නtan−1 𝑥 𝑑𝑥 .
Example
Evaluate
න𝑒 𝑥 sin 𝑥 𝑑𝑥 .
Example
Calculate
8
න𝑡 15 𝑒 −𝑡 𝑑𝑡 .
Example
Evaluate
නsin
𝑥 𝑑𝑥 .
Integration by Parts – Definite Integrals
𝑏
𝑏
𝑏
𝑎
𝑎
න 𝑢 𝑑𝑣 = 𝑢𝑣 ቚ − න 𝑣 𝑑𝑢
𝑎
Example
Find the area of the region bounded by the curve 𝑦 = 𝑥𝑒 −𝑥 and the 𝑥 −axis
from 𝑥 = 0 to 𝑥 = 4.
Example - Additional
Evaluate the integral
න𝑓 −1 𝑥 𝑑𝑥 .
Then, use the result to evaluate
නsec−1 𝑥 𝑑𝑥.
Example (Cont’d)