Integration by Parts

# Integration by Parts - October 1 2010 1 1 Integration by...

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October 1, 2010 1 1. Integration by parts: In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other (ideally simpler) integrals. The rule arises from the product rule of diﬀerentiation. The integration by parts states that ˆ udv = ˆ v du - uv (a) Diﬀerential parts du, dv : Suppose u is a function of x , then du = u Í ( x ) dx, or equivalently du dx = u Í ( x ) Similarly, for v . As an example, we could write e x dx = d ( e x ), e - x dx = d ( - e - x ), x 2 dx = d ( x 3 / 3). Note that ´ du = u for any function u . (b) Example: ´ xe x dx As we know, d ( e x ) = e x dx , therefore we can write this integral as ´ xd ( e x ). If we let u = x, v = e x , then using the integration by parts formula, we have ˆ xd ( e x ) = xe x - ˆ e x dx = xe x - e x . (c) Tabular integration by parts: The intergration by parts technique could directly result in the following re- cursive formula for calculating

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Integration by Parts - October 1 2010 1 1 Integration by...

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