Barrons AP Calculus

Parts   Formula

The Integration by Parts Formula stems from the equation for the derivative of a
product:

    or, or  more    conveniently    d(uv)   =   u dv + v du.

Hence, u dv = d(uv) − v du and integrating gives us , or

,

the Integration by Parts Formula. Success in using this important technique
depends on being able to separate a given integral into parts u and dv so that (a)
dv can be integrated, and (b) is no more difficult to calculate than the

original integral.
The following acronym may help to determine which function to designate as
u when using Integration by Parts. The acronym is LIPET. Choose u in this
order: (1) a Logarithmic function, (2) an Inverse trigonometric function, (3) a
Polynomial function, (4) an Exponential function, and (5) A Trigonometric
function. NOTE: Some may advise to swap the last two and always choose
Exponential last.

Example 43 __

Find .

SOLUTION: We let u = x and dv = cos x dx. Then du = dx and v = sin x. Thus,
the Parts Formula yields

.

Example 44 __

Find .

SOLUTION: We let u = x^2 and dv = ex dx. Then du = 2 x dx and v = ex, so

. We use the Parts Formula again, this time letting u = x