$\displaystyle \int x e^{7x^2+1}$
The problem is that you didn't put the dx in the first integral.
It's $\displaystyle \int xe^{7x^2+1} ~{\color{red}dx}$
So you can't think of the dx/dt...
$\displaystyle t=7x^2+1 \Rightarrow \frac{dt}{dx}=14x \Rightarrow dx=dt\cdot\frac{1}{14x}$
So you have to substitute $\displaystyle dx$ by $\displaystyle dt\cdot\frac{1}{14x}$
and the x will simplify : $\displaystyle \int xe^{7x^2+1} ~dx=\int x e^t \cdot \frac{1}{14x} ~dt=\frac{1}{14} \int e^t ~dt$
After that, don't forget to back substitute (you will have a formula with t, but t=7x^2+1, so you have to change it)