5.4. INTEGRATION BY PARTS
309
Exercises
1. Let f (t) = te −2t and F(x) =
x
0
f (t) dt.
(a) Determine F ′ (x).
(b) Use the First FTC to find a formula for F that does not involve an integral.
(c) Is F an increasing or decreasing function for x > 0? Why?
2. Consider the indefinite integral given by
e 2x cos(e x ) dx.
(a) Noting that e 2x = e x · e x , use the substitution z = e x to determine a new,
equivalent integral in the variable z.
(b) Evaluate the integral you found in (a) using an appropriate technique.
(c) How is the problem of evaluating
e 2x cos(e 2x ) dx different from evaluating
the integral in (a)? Do so.
(d) Evaluate each of the following integrals as well, keeping in mind the approach(es)
used earlier in this problem:
•
e 2x sin(e x ) dx
•
e 3x sin(e 3x ) dx
•
xe x 2 cos(e x 2 ) sin(e x 2 ) dx
3. For each of the following indefinite integrals, determine whether you would use usubstitution, integration by parts, neither*, or both to evaluate the integral. In each
case, write one sentence to explain your reasoning, and include a statement of any
substitutions used. (That is, if you decide in a problem to let u = e 3x , you should state
that, as well as that du = 3e 3x dx.) Finally, use your chosen approach to evaluate each
integral. (* one of the following problems does not have an elementary antiderivative
and you are not expected to actually evaluate this integral; this will correspond with a
choice of “neither” among those given.)
(a)
x 2 cos(x 3 ) dx
(b)
x 5 cos(x 3 ) dx (Hint: x 5 = x 2 · x 3 )
(c)
x ln(x 2 ) dx
(d)
sin(x 4 ) dx
(e)
x 3 sin(x 4 ) dx
(f)
x 7 sin(x 4 ) dx
309
Exercises
1. Let f (t) = te −2t and F(x) =
x
0
f (t) dt.
(a) Determine F ′ (x).
(b) Use the First FTC to find a formula for F that does not involve an integral.
(c) Is F an increasing or decreasing function for x > 0? Why?
2. Consider the indefinite integral given by
e 2x cos(e x ) dx.
(a) Noting that e 2x = e x · e x , use the substitution z = e x to determine a new,
equivalent integral in the variable z.
(b) Evaluate the integral you found in (a) using an appropriate technique.
(c) How is the problem of evaluating
e 2x cos(e 2x ) dx different from evaluating
the integral in (a)? Do so.
(d) Evaluate each of the following integrals as well, keeping in mind the approach(es)
used earlier in this problem:
•
e 2x sin(e x ) dx
•
e 3x sin(e 3x ) dx
•
xe x 2 cos(e x 2 ) sin(e x 2 ) dx
3. For each of the following indefinite integrals, determine whether you would use usubstitution, integration by parts, neither*, or both to evaluate the integral. In each
case, write one sentence to explain your reasoning, and include a statement of any
substitutions used. (That is, if you decide in a problem to let u = e 3x , you should state
that, as well as that du = 3e 3x dx.) Finally, use your chosen approach to evaluate each
integral. (* one of the following problems does not have an elementary antiderivative
and you are not expected to actually evaluate this integral; this will correspond with a
choice of “neither” among those given.)
(a)
x 2 cos(x 3 ) dx
(b)
x 5 cos(x 3 ) dx (Hint: x 5 = x 2 · x 3 )
(c)
x ln(x 2 ) dx
(d)
sin(x 4 ) dx
(e)
x 3 sin(x 4 ) dx
(f)
x 7 sin(x 4 ) dx