Quick question about changing limits when doing U-substitution.

Question

when doing U-substitution on definite integrals, is it possible to make the shortcut of not changing the limits if I change "u" back to "x" once the function has been integrated?

I believe I'm allowed to make this shortcut but want to check.

Thanks.

Answer 1

Yes, you can think of this as doing the indefinite integral $\int f(x)dx = F(x) + C$ by a u substitution (for which you need to to change u back to x), and then applying $$\int_a^b f(x)dx = F(b)-F(a).$$

However when writing out a computation of a definite integral with u substitions, it's good practice to change the limits at each intermediate step, so each line is technically correct.

Answer 2

You don't need to change limits.

e.g. integrating $∫8x(x^2+3)^3 \,dx$.

You can expand or do a u-substitution.

Either way, when you convert back from u to x, you have the same expression.

Therefore, no need to change limits.