1. ## U substitution

Quick question about u substitution. When do you know when you need to change coordinates? I remembered learning about this in my second course of calculus, and now i have to revisit it, but i forgot how it works. A example with a brief explanation would be appreciated, or any external inks as well. Thanks.

2. I was always told to change the terminals when the variable itself changed.

Consider the following problem

$\int_2^3\frac{2x}{x^2+1}~dx$

We can see that $2x$ is the derivative of $x^2+1$ therefore we make

$u = x^2+1 \Rightarrow \frac{du}{dx} = 2x$

As the terminals x = 2,3 are with respect to x and our substituted variable is u we use

$u = x^2+1$ to change them.

x=2 gives $u = 2^2+1 = 5$

x=3 gives $u = 3^2+1 = 10$

And now we have

$\int_5^{10}\frac{1}{u}\frac{du}{dx}~dx$

$\int_5^{10}\frac{1}{u}~du$