Intergral Help for calculus 
Here is what I have. Am I on the right track?

Hint:
First, let us find a antiderivative for $xe^{6x^2-9}$, by rewriting the integrand a little bit.
$$\int xe^{6x^2-9}\,dx=\frac{1}{12}\int12xe^{6x^2-9}\,dx$$
Now note that the $12x$ is exactly the derivative of $6x^2-9$, therefore the indefinite integral is equal to $\frac{1}{12}e^{6x^2-9}+C$.
It's up to you now to use the antiderivative to find the definite integral. I am sure you can take it from here.

Edit:
$$\int_1^4xe^{6x^2-9}\,dx=\left.\frac{1}{12}e^{6x^2-9}\right|_1^4=\frac{1}{12}e^{6(4)^2-9}-\frac{1}{12}e^{6(1)^2-9}=\frac{1}{12}\left(e^{87}-e^{-3}\right)$$
