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5.4. INTEGRATION BY PARTS
are differentiable functions of x, then
d
dx
[ f (x) · g(x)] = f (x) · g
′ (x) + g(x) · f
′ (x).
(a) For each of the following functions, use the Product Rule to find the function’s
derivative. Be sure to label each derivative by name (e.g., the derivative of g(x)
should be labeled g ′ (x)).
i. g(x) = x sin(x)
ii. h(x) = xe x
iii. p(x) = x ln(x)
iv. q(x) = x 2 cos(x)
v. r(x) = e x sin(x)
(b) Use your work in (a) to help you evaluate the following indefinite integrals. Use
differentiation to check your work.
i.
xe
x + e
x dx
ii.
e
x (sin(x) + cos(x)) dx
iii.
2x cos(x) − x
2 sin(x) dx
iv.
x cos(x) + sin(x) dx
v.
1 + ln(x) dx
(c) Observe that the examples in (b) work nicely because of the derivatives you
were asked to calculate in (a). Each integrand in (b) is precisely the result of
differentiating one of the products of basic functions found in (a). To see what
happens when an integrand is still a product but not necessarily the result of
differentiating an elementary product, we consider how to evaluate
x cos(x) dx.