Answers and comments for Math 242 Homework 7.


#1. Final answer is 0. 

They can get away with saying (M-4)/(M-3)
"obviously" approaches 1. I don't require
that they use L'Hopital's rule.


#2. Final answer is 7/12.

Intersection points should be -1/2 and 1/2.
Because of symmetry, they can do the "shortcut"
of integrating from 0 to 1/2, then doubling.


#3. Final answer can be written various ways:

5 - 10ln(3/2)

5 - 10ln(15/10)

5 - 10ln(3) + 10ln(2)

They can do substitution, or algebraic rearranging.

(Numerical answer is approx 0.945 but they don't
need to include it.)


#4. Final answer is ln(2).

To find the antiderivative, they should use partial
fractions. Antiderivative should be

ln(x-4) - ln(x-3)  or  ln( (x-4)/(x-3) ).

Then, to evaluate the integral, they may use question 1.


#5. Final answer is

(x/2)*(sin(ln x)-cos(ln x))

There may be different ways to do this. They can start
with a substitution, or they can start with integration
by parts.

Probably, they have to integrate by parts twice.


#6. Final answer is

ln(3) - 1/2

which is approximately 0.5986 (they don't need the
numerical value).

Along the way, their steps should be

y = ln(1-x^2)

y' = -2x/(1-x^2)

(y')^2 = 4x^2/(1-x^2)^2

1+(y')^2 = ... = (1+x^2)^2 / (1-x^2)^2

sqrt(1+(y')^2) = (1+x^2) / (1-x^2)

So then they have to integrate (1+x^2)/(1-x^2)
from x=0 to x=1/2. They should use partial fractions.


#7. Final answer is: it converges to 1/9.

They should do integration by parts (and only need to
do it once).


#8. Final answer is: it diverges.

Probably, they will do the substitution u = ln(x).