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Do Now – #1 and 2, Quick Review, p.328 Find dy/dx: 1. 2.

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SECTION 6.3A Integration by Parts

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To “integrate by parts,” we first need to investigate the… Product Rule in Integral Form If u and v are differentiable functions of x, the Product Rule for differentiation gives: Next, let’s rearrange the terms a bit:

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To “integrate by parts,” we first need to investigate the… Product Rule in Integral Form Now, integrate both sides with respect to x:

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To “integrate by parts,” we first need to investigate the… Product Rule in Integral Form Writing this equation in simpler differential notation yields Integration by Parts Formula: the Integration by Parts Formula:

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Integration by Parts Formula This formula expresses one integral in terms of a second integral. With proper choices for u and v, this second integral will be easier to evaluate…a very useful technique (but note: it doesn’t always work…)

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Integration by Parts Formula A bit of wisdom from your textbook’s authors: We want u to be something that simplifies when differentiated. We want v to be something “managable” when integrated. L I P E T When choosing u, use L I P E T for order preference: Natural Logarithm (L) Inverse Trig Function (I) Polynomial (P) Exponential (E) Trig Function (T)

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Guided Practice Evaluate Keep the formula in mind!!! Let Then Let Then The original integral is transformed:

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Guided Practice Evaluate What if we had made different choices for u and v? Let A poor choice, since we still don’t know how to integrate dv to obtain v… Let

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Guided Practice Evaluate What if we had made different choices for u and v? Let The new integral: Let Then …is worse than the original!!!

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Guided Practice Evaluate What if we had made different choices for u and v? Let The new integral: Let Then …is also a stinker!!!

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Guided Practice Evaluate Now, differentiate to confirm your answer!

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Guided Practice Evaluate Let Now we integrate by parts with this new integral!!! This is an example of repeated use of I.B.P…

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Guided Practice Evaluate Let

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Guided Practice Evaluate This technique works with integrals in the form in which f can be differentiated repeatedly to zero, and g can be integrated repeatedly without difficulty…

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f(x) and its derivatives An alternative to the long-cut is to use tabular integration in such situations: g(x) and its integrals ( + ) ( – ) ( + )

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Guided Practice Evaluate: Tabular integration: f(x) and its derivativesg(x) and its integrals (+) (–)

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