questions on substitution

See my notes below.
When you respond to questions, insert your response into the questions. It's can be a hassle for me, and will usually be even more of one for you, to have to go back to a different document to see what the questions were. It wasn't a real problem for me, because you did include the function, but I did have to go back and see exactly what I had asked you for each one.
Insert my comments below into my original questions so we can maintain the thread on these integrals. My questions on each were
* What are your options for u substitution (there may be just one option, or there may be more than one; list all possible options whether they ultimately help you solve the integral or not)?
* In each case, what would be du?
You went one step further and attempted to express the integral in terms of the substitution. Generally you had the right idea but you didn't quite get it right. I've inserted comments and subsequent questions which should help straighten you out. My questions ask you to check out all possibilities. It is important to see why some possibilities work, some don't, and also why many integrals just can't be done by substitution.
Insert your answers into this document and send the document to me.
If you have questions on some of the graphs, please send them also.

If you are integrating x e^(x^2) with respect to x,

u = x^2 du = 2x dx so 1/2 du = x dx

Is x dx present in the original integral? If so (and of course it is) then you can substitute 1/2 du for x dx. You also substitute u for x^2.
What therefore is your integrand in terms of u and du?
What is the integral of this expression?
If you substitute x^2 for u in your integral what do you get?

u = x e^(x^2) is also a possible substitution. What would this give for du?
Can you find this du in the original integral?
If not, then the substitution u = x e^(x^2) doesn't work and you can abandon it.

Then this leaves us with 1/2 e^(x^2)

Not so. x e^(x^2) is not the same thing as 1/2 e^(x^2), since x is not generally equal to 1/2.

If you are integrating cos(x^3) e^(x^2) with respect to x,

u = x^3 du = 3x^2 so 1/3 du = x^2 dx

Is x^2 dx present in the original integral?
If not, then you can't go any further with this substitution.
If so then you can substitute 1/3 du for x^2 dx, and you also substitute u for x^3. If this is so then you can continue with the following questions; if it isn't so then you should already have abandoned this possibility:
What therefore is your integrand in terms of u and du?
What is the integral of this expression?
If you substitute x^3 for u in your integral what do you get?

w = x^2 dw = 2x dx so 1/2 dw = x dx

Is x dx present in the original integral?
If not, then you can't go any further with this substitution.
If so then you can substitute 1/2 dw for x dx, and you also substitute u for x^2. If this is so then you can continue with the following questions; if it isn't so then you should already have abandoned this possibility:
What therefore is your integrand in terms of u and du?
What is the integral of this expression?
If you substitute x^2 for u in your integral what do you get?

1/6 cos(x^3) e^(x^2)

You could also have tried u = cos(x^3), or u = e^(x^2).
What is du for each of these possibilities?
Why doesn't either of these possibilities work?

If you are integrating sin(x) e^(cos(x)) with respect to x,

u = cos(x) du = sin(x)

Is sin(x) dx present in the original integral?
If not, then you can't go any further with this substitution.
If so then you can substitute du for sin(x) dx, and you also substitute u for cos(x). If this is so then you can continue with the following questions; if it isn't so then you should already have abandoned this possibility:
What therefore is your integrand in terms of u and du?
What is the integral of this expression?
If you substitute cos(x) for u in your integral what do you get?

du = 1 dx

This isn't so. du = sin(x) dx. sin(x) isn't equal to the constant number 1.

e^(cos(x))

If you are integrating (x^2 + 3x + 2) sin(x) with respect to x,

u = x^2 + 3x + 2 du = 2x + 3 du" = 2

You don't bother with a second derivative of u.

Is (2x+3) dx present in the original integral?
If not, then you can't go any further with this substitution.
If so then you can substitute du for (2x+3) dx, and you also substitute u for (x^2+3x+2). If this is so then you can continue with the following questions; if it isn't so then you should already have abandoned this possibility:
What therefore is your integrand in terms of u and du?
What is the integral of this expression?
If you substitute x^2+3x+2 for u in your integral what do you get?

so [(x^2 + 3x + 2)sin(x) - (2x + 3) cos (x) + 2 sin(x)] + C

you wouldn't get this--nothing this complicated here

If you are integrating x^2 ln(x^3) with respect to x,

u = x^3 du = 3x^2 so 1/3 du = x^2

du = 3 x^2 dx, not just 3 x^2.
Is x^2 dx present in the original integral?
If not, then you can't go any further with this substitution.
If so then you can substitute 1/3 du for x^2 dx, and you also substitute u for x^3. If this is so then you can continue with the following questions; if it isn't so then you should already have abandoned this possibility:
What therefore is your integrand in terms of u and du?
What is the integral of this expression?
If you substitute x^2 for u in your integral what do you get?

Then

1/3 ln(x^3)

x^2 ln(x^3) is not the same as 1/3 ln(x^3), since x^2 is not the constant number 1/3.

Note also that u = x^2 is a possibility, though it doesn't work out.
So is ln(x^3). What would be du if u = ln(x^3)? This doesn't work out here, but there are integrals where you would use this. For example if the integral was ln(x^3) / x, the substitution u = ln(x^3) would work.

If you are integrating (x - 3) / (x + 3) ^ 2 with respect to x,

I have no clue on this one I have looked but this is were I dont understand.

No substitution will work here, but you still need to try the possibilities , which are
u = x-3
u = x+3
u = 1/(x+3)
u = (x+3)^2
u = 1/(x+3)^2.

If you are integrating (2x + 4) / ( x^2 + 4 x + 3) with respect to x,

2(x+2)/[(x+2)^2 -1]

I have gotten this fair with this problem but I dont know how to go from here. I have been working on these two problem the past few nights and figured I would just quit. Next time I will get back in a faster manner. I forgot about another test I had this week too thats why I did not get back as fast as I normally do. I do have some question on graph problems that are also on the test as well.

questions on substitution

Is x dx present in the original integral? If so (and of course it is) then you can substitute 1/2 du for x dx. You also substitute u for x^2. What therefore is your integrand in terms of u and du? What is the integral of this expression? If you substitute x^2 for u in your integral what do you get?

u = x e^(x^2) is also a possible substitution. What would this give for du? Can you find this du in the original integral? If not, then the substitution u = x e^(x^2) doesn't work and you can abandon it.

Not so. x e^(x^2) is not the same thing as 1/2 e^(x^2), since x is not generally equal to 1/2.

You could also have tried u = cos(x^3), or u = e^(x^2). What is du for each of these possibilities? Why doesn't either of these possibilities work?

This isn't so. du = sin(x) dx. sin(x) isn't equal to the constant number 1.

You don't bother with a second derivative of u.

you wouldn't get this--nothing this complicated here

x^2 ln(x^3) is not the same as 1/3 ln(x^3), since x^2 is not the constant number 1/3.

Note also that u = x^2 is a possibility, though it doesn't work out. So is ln(x^3). What would be du if u = ln(x^3)? This doesn't work out here, but there are integrals where you would use this. For example if the integral was ln(x^3) / x, the substitution u = ln(x^3) would work.

No substitution will work here, but you still need to try the possibilities , which are u = x-3 u = x+3 u = 1/(x+3) u = (x+3)^2 u = 1/(x+3)^2.

Your possibilities are u = x+2 u = x^2 + 4x + 3 u = 1/(x^2 + 4x + 3) u = x^2 + 4x. Yu should check out each possibility.

Is x dx present in the original integral? If so (and of course it is) then you can substitute 1/2 du for x dx. You also substitute u for x^2.
What therefore is your integrand in terms of u and du?
What is the integral of this expression?
If you substitute x^2 for u in your integral what do you get?

u = x e^(x^2) is also a possible substitution. What would this give for du?
Can you find this du in the original integral?
If not, then the substitution u = x e^(x^2) doesn't work and you can abandon it.

You could also have tried u = cos(x^3), or u = e^(x^2).
What is du for each of these possibilities?
Why doesn't either of these possibilities work?

Note also that u = x^2 is a possibility, though it doesn't work out.
So is ln(x^3). What would be du if u = ln(x^3)? This doesn't work out here, but there are integrals where you would use this. For example if the integral was ln(x^3) / x, the substitution u = ln(x^3) would work.

No substitution will work here, but you still need to try the possibilities , which are
u = x-3
u = x+3
u = 1/(x+3)
u = (x+3)^2
u = 1/(x+3)^2.

Your possibilities are
u = x+2
u = x^2 + 4x + 3
u = 1/(x^2 + 4x + 3)
u = x^2 + 4x.
Yu should check out each possibility.