Integration+by+Parts

This method of integration is to be used when there is a product rule in effect and you have to take the integral of it. The formula for the method of integration by parts is given by.
 * Integration by Parts**

Choosing what will be your u value and dv value is very important. If //u// and //v// are functions of //x//, the [|product rule for differentiation] that we met earlier gives us: > Rearranging, we have: > Integrating throughout, with respect to //x//, we obtain the formula for **integration by parts:** > This formula allows us to turn a complicated integral into more simple ones. We must make sure we choose //u// and //dv// carefully. Function //u// is chosen so that is **simpler** than //u//.

Substituting into the integration by parts formula, we get: >

> **Example problem:** > **Evaluate** > First let us point out that we have a definite integral. Therefore the final answer will be a number not a function of x! Since the derivative or the integral of lead to the same function, it will not matter whether we do one operation or the other. Therefore, we concentrate on the other function. Clearly, if we integrate we will increase the power. This suggests that we should differentiate and integrate. Hence > > After integration and differentiation, we get > > The integration by parts formula gives > > It is clear that the new integral is not easily obtainable. Due to its similarity with the initial integral, we will use integration by parts for a second time. The same discussion as before leads to > > which implies > > The integration by parts formula gives > > Since, we get > > which finally implies > > Easy calculations give >

**Work Cited**

Bourne, Murray. "Integration by Parts." //Interactive Mathematics//. 8 May 2011. Web. 24 May 2011. .

Khamsi, Muhamed A. "Integration by Parts: Example 2." //S.O.S Mathmatics//. Math Medics. Web. 24 May 2011.

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