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

SECTION 6.3A Integration by Parts

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:

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

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:

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…)

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)

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

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

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!!!

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

Guided Practice Evaluate Now, differentiate to confirm your answer!

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

Guided Practice Evaluate Let

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…

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

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