Page 211 - Calculus Demystified

198 or CHAPTER 7 Methods of Integration

u · v dx = u · v − v · u dx.

It is traditional to abbreviate u (x) dx = du and v (x) dx = dv. Thus the
integration by parts formula becomes

udv = uv − vdu.

Let us now learn how to use this simple new formula.

EXAMPLE 7.1
Calculate

x · cos xdx.

SOLUTION FLY
We observe that the integrand is a product. Let us use the integration by parts
TEAM
formula by setting u(x) = x and dv = cos xdx. Then

u(x) = x du = u (x) dx = 1 dx = dx
v(x) = sin x dv = v (x) dx = cos xdx

Of course we calculate v by anti-differentiation.
According to the integration by parts formula,

x · cos xdx = udv

= u · v − vdu

= x · sin x − sin xdx

= x · sin x − (− cos x) + C
= x · sin x + cos x + C.

Math Note: Observe that we can check the answer in the last example just by
differentiation:
d
dx [x · sin x + cos x + C]= 1 · sin x + x · cos x − sin x = x · cos x.
The choice of u and v in the integration by parts technique is significant. We
selected u to be x because then du will be 1 dx, thereby simplifying the integral.

Team-Fly
®