Barrons AP Calculus - David Bock

D. INTEGRATION BY PARTS

Parts Formula

The 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 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) du is no more difficult to
calculate than the original integral.

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 = 2x dx and v = ex, so

We use the Parts Formula again, this time letting u = x and dv = ex dx so that du = dx and v = ex.

Thus,

EXAMPLE 45

Find I =

SOLUTION: To integrate, we can let u = ex and dv = cos x dx; then du = ex dx, v = sin x. Thus,

To evaluate the integral on the right, again we let u = ex, dv = sin x dx, so that du = ex dx and v =