Computing calculus limits often involves using an algebraic trick. In this post we’ll look at the example of a ratio of two polynomials. In this case you’ll want to factor the polynomials so that you can eliminate one of the troubling terms. Here’s an example:
If you just set x = 2, you get 0/0. However [...]
Posted on February 7th, 2010 by AdamP
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Integration by parts with limits is really the same as integration by parts with indefinite integrals until the very end of your calculation. Let’s see how to do integration by parts with limits with an example:
Its always a good idea to start out writing the integration by parts formula:
For now, we can simply ignore [...]
Posted on February 6th, 2010 by AdamP
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In some cases, integration by parts has to be used with another technique like u-substitution. For example lets do
Its always a good idea to start out writing the integration by parts formula:
Since taking the derivative of x would just give 1, we’ll take that to be u. So
This integral must be done using u-substitution. [...]
Posted on February 6th, 2010 by AdamP
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Integration by u-substitution is a technique that helps us evaluate many integrals that can’t be done using elementary techniques. That is, we know that several “basic” integrals can be written down by inspection once you learn what the rules are. For example using the power rule you know that:
But what if you had:
This type of [...]
Posted on February 5th, 2010 by AdamP
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Integration by parts can be used to integrate the product of a trig function with an exponential. For example, consider:
First let’s recall the integration by parts formula:
The trick is to pick u and v in such a way that
is a simpler expression. In this case, try
Then
(ignore constants of integration for now, we’ll add it [...]
Posted on February 2nd, 2010 by AdamP
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