How Do You Solve These Integral Problems Using U-Substitution?

[EasyStyle4747]

Ok, ill try to explain what i did:

1.
integral of [square root of( e^t-3)dt]

Sorry i didn't know how to do the square roots symbol and the integral symbol. Anyways, I tried to set e^t-3 as u and then got dt=du/e^t. Then I plugged in dt and couldn't go on after that.

2. integral of x^2[e^(x^3)]dx

so uh... u substitution? but then what?

Ok basically, i think i don't know what your supposed do when u need to do a u-sub with e^something. Your help would be most appreciated.

 

[HallsofIvy]

1. I don't have any problem with the square root sign but is that e^(t-3) or
(e^t)-3?
If it is e^(t-3) then sqrt(e^(t-3))= (e^(t-3))^(1/2)= e^((t-3)/2). Let u= (t-3)/2 so
du= (1/2)dt or dt= 2du. The integral becomes 2 integral e^u du.

2. Yes, a substitution- seeing that "complicated" x^3 in the exponent and x^2 multiplying, you should immediately think of u= x^3 (NOT u= e^something- the e is not the problem!). Then du= 3 x^2 dx or (1/3)du= x^2 dx so the integral becomes
(1/3) integral e^u du.

 

[Data]

Well, thinking of $$u = e^{x^3}$$ wouldn't hurt in this case regardless

 

[EasyStyle4747]

thnx, i see how the second one works now.

For the first one, its actually square root of (e^t)-3. Sorry i didnt make that clear.

 

[dextercioby]

Then make the substitution
$$e^{t}=u$$
,then the substitution
$$u=v^{2}+3$$

Daniel.