1. ## Integration of exponential

$\displaystyle \int x e^{7x^2+1}$

2. Hello,

7x^2+1 is a polynomial of degree 2.
x is a polynomial of degree 2-1.

So you have to think : substitution !

-> substitute $\displaystyle t=7x^2+1$

3. So what happens after I substitute?

It becomes $\displaystyle \int x e^t$?

or

$\displaystyle \int \sqrt{\frac{t-1}{7}} e^t$?

4. Originally Posted by nerdzor
So what happens after I substitute?

It becomes $\displaystyle \int x e^t$?

or

$\displaystyle \int \sqrt{\frac{t-1}{7}} e^t$?
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)