Evaluating definite integral U substitution

Question

$$\int_0^1 {2x(x^2+1)^4dx} $$

I u-substituted $ x^2+1 $, found the anti derivative, and plugged in the original limits ($0$ and $1$) into the substituted equation to get new limits ($1$ and $2$). The answer I got is $\frac{3093}5$.

I then checked on the calculator to see the correct answer is $\frac{31}5$.

So I plugged in the original limits into the same FTC anti derivative thing and got the correct answer this time. Can someone explain why? I thought when you use u-substitution, you need to plug in the original limits into the substituted equation and use the new limits.

Answer 1

Let $$u=x^2+1$$

$$\frac{du}{dx}=2x$$

$$\int_0^1 {2x(x^2+1)^4dx} = \int_1^2 u^4 du = \frac{2^5-1^5}{5}=\frac{31}{5}$$