Definite Integration using Power Substitution

Question

Given the definite integral:

$$\int_{1}^{2}\left(x\sqrt{x+3}\right)\text{d}x$$

We can make the Power Substitution: $$\begin{align} u^2=&&x+3 \\ 2u\text{d}u=&&\text {d}x \end{align}$$

We get the following: (without the limits)

$$\int{\left(\left(u^2-3\right)\times u\times 2u\right)\text{d}u}$$

(Yes I know, there is a much simpler way to go about this.)
However, when we change the bounds of integration we are left with this upper limit for example: $$\begin{align} u&&=\pm\sqrt{x+3} \\ &&=\pm\sqrt{2+3} \\ &&= \pm\sqrt{5} \end{align}$$

Should we use $+\sqrt{5}$ or $-\sqrt{5}$, How do we know which value to use?

I assume it depends on the integrand function's domain. But what about the case where both values satisfy the domain?

Answer 1

The problem you have is that the map $u \mapsto x$ is not one-to-one (injective). This is discussed with some complexity here. There are two intervals in $u$ that are mapped to the same interval in $x$: $[2,\sqrt{5}]$ and $[-\sqrt{5}, -2]$. You have two choices:

• Pick one interval, either $\int_2^\sqrt{5} \dots \,\mathrm{d}u$ or $\int_{-\sqrt{5}}^{-2} \dots \,\mathrm{d}u$, or
• Use both intervals $\frac{1}{2} \left( \int_2^\sqrt{5} \dots \,\mathrm{d}u + \int_{-\sqrt{5}}^{-2} \dots \,\mathrm{d}u \right)$ and realize you're getting two copies of the result, so you divide by two.

You know your $x$ interval, $[1,2]$, is connected and the two intervals we got are not connected to each other, so either is fine. This can be much more complicated...

Answer 2

It does not matter, as long as we are consistent. We can either integrate from $u=2$ to $u=\sqrt{5}$ or from $u=-2$ to $u=-\sqrt{5}$.