Chapter 5.05: Integration by Substitution

Remember to check your answers by differentiating, when applicable.

Remember to avoid some common mistakes:
* still writing the integral sign after taking the antiderivative
* dropping the integral sign without actually taking the antiderivative
* plugging in x=0 for e^x or cos(x) and getting 0 instead of 1
* mixing "u" and "x" in the same integral expression
* forgetting to write "u" values as the limits instead of "x" values,
 when you switch from x to u.

Drill problems:
#1
#4
#8
#10
#14
#15
#16
#17
#18
#20
#26; do this one from scratch; don't use a "Big 4" formula on it.
#28

#53
#56
#60
#64
#72 WEP; use Wolfram Alpha to simplify near the end
#76 WEP
#77 WEP

Applied problems:
#80  basal metabolism rate like an offset cosine
For this one, also comment on the
plausibility of the R(t) function given.

#83 WEP; respiratory cycle: I like the basic idea of this problem, but it
is poorly phrased. It doesn't say what the initial volume of air in the
lungs is. It does say to consider only the "inhaled" air, so we could
say that the initial volume of _inhaled_ air is 0 (other air sitting in
the lungs wouldn't count as inhaled during this cycle).

#81
#82
#84; be careful to find the right starting and stopping values for t.
Hint: it isn't t=3 to t=4, and it isn't t=3 to t=5.
#86 WEP
#88 WEP

In the following problems, int_a^b means the integral from a to b.

Question A: WEP (similar to 88) suppose that b>0, and f(x) is continuous. Is
int_0^b f(x) dx 
equal to
int_0^b f(b-x) dx?  Draw diagrams and discuss. 
Make up an example f(x) and work it through.

Question B: Let r(t) = the arrival rate (in cars/hour) at time "t".
Let A(t) = the total number of cars that arrived between midnight (time 0)
and now (time "t").  What do you think of the following argument?
A(t) = int_0^t r(t) dt
So, if we want to know the number that have arrived up to time 10am,
we should do   A(10) = int_0^10 r(10) dt ?

Question C: This one counts as a Drill problem.
Find these definite integrals
i) int_-2^+2 x*(7x^2)^5 dx (use u-substitution, and don't take advantage of
symmetry)
ii) int_0^5 exp(-0.2 t) dt (it's important in probability applications)
iii) int_0^1000 exp(-0.2 t) dt (same function as part (ii), but different upper
limit)
iv) what value of "b" solves int_0^b exp(-t) dt = 0.5 ?

Question D: This one counts as a Drill problem.
i) Graph f(x) = 1/(1+exp(x)) from x=-3 to +3
ii) Find the antiderivative.  Use the fact that
1/(u*(u+1)) = (1/u) - (1/(u+1))
iii) Graph the antiderivative, and make sure it is reasonably the
antiderivative of f(x)

Question E: This one counts as a Drill problem.
i) Graph f(x) = exp(x) / (1+exp(x))^2 from x=-3 to +3
ii) Find the antiderivative. 
iii) Graph the antiderivative, and make sure it is reasonably the
antiderivative of f(x)
iv) What is the int_-1^+1 exp(x)/(1+exp(x))^2 dx ?

Question F: For people who are going on to Calc II.
Do a u-substitution for each of these problems, then stop as you realize
that didn't help solve it at all.  Say out loud: "I guess I'll just have
to wait until Chapter 7.3 to solve that one!" for each.  Do NOT use any
trig identities like   sin^2+cos^2=1.
i) int sqrt(1- (sin(x))^2) cos(x) dx
ii) int sqrt(1- (cos(x))^2) sin(x) dx
iii) int 1/sqrt(1- (sin(x))^2) cos(x) dx
iv) int sqrt(1+ (sinh(x))^2) cosh(x) dx [sinh is "hyperbolic sine", and the
deriv. of sinh is cosh]


Question G: (optional)
This question is about low-pass filtering: removing the high-frequency
components of a signal. Here we use a moving average.
i) Let L, a, t, and p be arbitrary constants.  Compute
(1/L) * integral from h=-L to 0 of sin(a*(t+h)+p) dh
[use the identity: cos(A) - cos(B) = -2*sin( (A+B)/2 ) * sin( (A-B)/2 ) 
and then note that one of those sine terms is constant with respect to t]
ii) Graph the amplitude of the result as a function of the frequency "a",
using p=0 and L=0.4
iii) Graph the log10(amplitude) of the result as a function of log10(a)

------------ Big 4 Formulas: Possibly Extra Drill -----------------------
Check with me to see if these are required for you.

Recall our big 4 formulas:
integral of sin(ax+b) dx = (1/a) * -cos(ax+b) + C
integral of cos(ax+b) dx = (1/a) * sin(ax+b) + C
integral of exp(ax+b) dx = (1/a) * exp(ax+b) + C
integral of  1/(ax+b) dx = (1/a) * ln(abs(ax+b)) + C

On each of the following, use a Big 4 Formula to do it in one step,
avoiding doing a u-substitution.  But watch out!  Some of these
do not work with the Big 4 Formulas.  Find them and say which ones they
are, and why the Big 4 don't apply--you don't have to solve them.

integral of cos(3t) dt
integral of exp(-0.3 x) dx
integral of 1/(3+11x) dx
integral of sin(440s+pi/6) ds
integral of exp(-x/5) dx
integral of 155 cos(2*pi*t) dt
integral of sin(2*pi*(t-6)/24) dt
integral of exp(7x^2) dx
integral of exp(5(3-x)/8) dx
integral of 1/(1+3x^2) dx
integral of 5/(0.4 x+7) dx
integral of 9*sin(5x) dx
integral of t*cos(3t+2) dt