Changing the limits on Integrals

[kLPantera]

I'm confused as to when to change the limits on a definite integral.

Ex. Integral with the limits a=1, b=5, 3/(x+1)dx

I set u = x+1 and du = dx

I used u-substitution and everything worked out fine.

However for this one...

Ex. Integral with the limits a = 0, b = 2, 6x^2/sqrt((x^3)-1)

I used u-substitution u = sqrt((x^3)-1) and so 2du = (6x^2)dx
However the book says I need to change the limits of the integral.

So I'm not sure when to change the limits of an integral. Can anyone help? Thanks =D

 

[Dr. Seafood]

For the first integral, did you change the integration limits to [2, 6]?

For the second one, of course you must change the limits of the integral when you make a substitution, but also note that this is an improper integral since the integrand is discontinuous somewhere on [0, 2].

 

[kLPantera]

For the first integral no I did not.

 

[I like Serena]

Hi kLPantera!

You're integrating with respect to x from a to b.
When you substitute u=u(x) you change the expression to read "du" instead of "dx".
This means that exactly from this moment on you're integrating with respect to u.
The limits for u=u(x) are then u(a) and u(b).

 

[kLPantera]

Ah I understand it now lol