Let’s say for example you want to evaluate this integral:
$$\int_0^{\pi/2} \sin(x)\cos(x)\,dx$$
The best way to do that is to use a substitution, namely you will let $u=\sin(x) \implies du=\cos(x)dx$, And you have to change the bounds of integration, so the integral will look like this :
$$\int_0^1 u\,du=\frac{1}{2}$$
But there is a way to do this integral without changing the bounds of integration:
$$\int_0^{\pi/2} u\,du=\frac{u^2}{2} \bigg\rvert_0^{\pi/2} $$
Now instead of calculating it directly you should go back to the original variable, namely $u=\sin(x)$ :
$$\frac{u^2}{2}\bigg\rvert_0^{\pi/2} = \frac{\sin^2(x)}{2} \bigg\rvert_0^{\pi/2}=\frac{1}{2} $$
My question is which is better changing the bounds of integration first or doing this method? I think that this method is quite nice because if you have a hard substitution and you don’t know how to solve for $u$ To get the new bounds, you can just use this method. Is that correct?
Best Answer
The reason why you wont see your second method (not changing the bounds) presented as often is that it is not correct to say
$$ \int_0^{\pi/2}\sin(x)\cos(x)\,dx = \int_0^{\pi/2} u \,du$$ while it is correct to say
$$ \int_0^{\pi/2}\sin(x)\cos(x)\,dx = \int_0^{1} u \,du.$$
Note that on the right hand side, $u$ is a dummy variable of integration and the substitution made before is irrelevant. One could just as easily write $\int_0^{\pi/2}v \,dv$ and the meaning doesn't change. So if you want to give a clear demonstration of your work given by a chain of equalities relating one quantity to the next, the 2nd method makes it difficult to do that.
In other words, there is no concise mathematical notation for "then go back and substitute $\sin(x)$ for $u$."
For working mathematicians, the two methods are `the same' (no difference in intellectual content) and therefore they are more likely to defer to the one that can be cleanly and correctly written down, even if they might still work it out in their mind using the second method.
So there really is a difference in the way people write vs how they think, especially in publications like textbooks with high editorial standards.
You should continue to think however you like, but make sure to use correct statements and equations in your writing.