Am I on the right track? Integration by substitution: (1/2) int e^x (e^x - 5)(1 + 2e^x)^8 2e^x dx

[HATLEY1997]

Am I on the right track? Integration by substitution

 

[Dr.Peterson]

There are a couple things that I can't follow:



Why is the e^x still there at the start of the first line after you moved it to become part of du, and why does it disappear later? How do you get from e^x - 5 to (u-1)/2, if I'm reading that right?

 

[HATLEY1997]

There are a couple things that I can't follow:

View attachment 37256

Why is the e^x still there at the start of the first line after you moved it to become part of du, and why does it disappear later? How do you get from e^x - 5 to (u-1)/2, if I'm reading that right?

I don’t even know myself what I have done looking back now!!!!

 

[HATLEY1997]

I don’t even know myself what I have done looking back now!!!!

There are a couple things that I can't follow:

View attachment 37256

Why is the e^x still there at the start of the first line after you moved it to become part of du, and why does it disappear later? How do you get from e^x - 5 to (u-1)/2, if I'm reading that right?

When I’ve rearranged the equation in part 2/3 was this incorrect?

 

[Dr.Peterson]

When I’ve rearranged the equation in part 2/3 was this incorrect?

I hadn't noticed step 3. You seem to have forgotten to subtract 5! That's probably the only actual error, though e^x disappearing may have been just luck. (I try never to write an integrand containing both old and new variables, though some people do.)

 

[HATLEY1997]

I hadn't noticed step 3. You seem to have forgotten to subtract 5! That's probably the only actual error, though e^x disappearing may have been just luck. (I try never to write an integrand containing both old and new variables, though some people do.)

 

[HATLEY1997]

I hadn't noticed step 3. You seem to have forgotten to subtract 5! That's probably the only actual error, though e^x disappearing may have been just luck. (I try never to write an integrand containing both old and new variables, though some people do.)

Ahh okay. So should step 3 be e^x=1/2(u-1)-5?

At what stage should the e^x be removed? I am getting confused between the e^x at the start and the e^x-5.

 

[Dr.Peterson]

Ahh okay. So should step 3 be e^x=1/2(u-1)-5?

No; why would you change what e^x is?

Rather, step 3 should say e^x - 5 = 1/2 (u-1) + 5.

Surely that's what you intended; but you have to write what you mean.

At what stage should the e^x be removed? I am getting confused between the e^x at the start and the e^x-5.

Please tell us where the "2e^x" at the end of the line came from. I assume you didn't just put it there because you wanted it, without checking how it relates to the line before.

The e^x at the start shouldn't need to be removed, It shouldn't be there at all, because it should have been moved.

 

[Steven G]

On your third line you had e^x. On the four line it just disappeared. As I said before, each and every piece of the integral must be accounted for.