Integration By Parts



Integration By Parts

We use this method when substitution is not able to be used and you have the product of two functions. One of the functions must be able to be integrated and we must be able to find the derivative of the other.

Some examples of when this would be used are; [pic]; [pic]; [pic]

The formula is this:[pic]

The simple approach is that we define one term as u and the other as dv. (The derivative of v). Once we define one term as u and the other of dv it is a simple matter of “plug and chug”. We find the derivative of u (to get du) and integral of dv (to get v) and than the integral of the product vdu and plug them into the formula.

Things to be careful of:

• Make sure that the term you are taking the derivative of can eventually become zero. This is important because the integral of the product vdu may require you to repeat the process of integration by parts again. If this is the case and your derivative does not eventually = 0 you could never complete the problem.

• When we choose u and v, choose u so that it is the easier of the terms to get the derivative of.

• Whenever we have lnx, we use that as u.(This is because we cannot integrate lnx)

Why this formula, [pic], works;

If we have [pic], where we would use the product rule to derivate, we would have;

[pic] (or uv’ + vu’), if we integrate both sides we get

[pic] ( [pic], we can rearrange these to become;

[pic]

(Notice that the [pic]. This is due to the integral “undoing” the derivative and leaving us with uv.

Example 1) [pic]

[pic]

[pic]

Example 2) [pic]

[pic]

Example 3) [pic]

[pic]

Example 4) [pic]

[pic]

OR

Use your calculator to integrate from 1 to e

[pic]

OR

[pic]

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