Example: limit (ln x)^2 dx | Lecture Note - Edubirdie

University:
Massachusetts Institute of Technology
Course:
18.01SC | Single Variable Calculus
Academic year:

2019
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207

Pages:

1
Author:

choirchik856

Example: � (ln x)2 dx To ﬁnish learning the method of integration by parts we just need a lot of practice. To this end, we’ll do two slightly more complicated examples. To integrate: � (ln x)2 dx, assign: u� = 2(ln x) u = (ln x)2 1 x v � = 1. v=x When we diﬀerentiate u we get something simpler, which is a good start. Plug ging u and v in to the formula for integration by parts we get: � � 1 2 2 (ln x) dx = (ln x) · x − 2 ln x �� x dx � �� x � �� uv � v uv = x(ln x)2 − 2 u� � ln x dx. We haven’t solved � the problem, but we’re back to the previous case; we recently computed that ln x dx = x ln x − x + c. So we have: � (ln x)2 dx = x(ln x)2 − 2 (x ln x − x) +c. � � �� ln x dx As we’ll see in the next example, this is typical. Integration by parts fre quently involves replacing a “hard” integral by an easier one.

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