I am having A LOT of problems with this one equation, could anyone help me? I know the answer, I just don't understand how to get there.
$$\int x^3 e^{x^2} dx$$
There's the equation, and the answer is: $$\int e^{x^2} x^3 dx = \frac 1 2 e^{x^2} (x^2 - 1) + \text{ constant} $$
I keep trying using the usual integration by parts method but I just can't get there no matter what. I use $$\int u dv = uv - \int v du.$$ but I seem to never get it right.
On
Let $u=x^2$ and $dv=xe^{x^2}\,dx$. Then $du=2x\,dx$ and $v$ can be taken to be $\frac{1}{2}e^{x^2}$. So we arrive at $$\frac{1}{2}x^2e^{x^2} -\int xe^{x^2}\,dx.$$ This last integral is straightforward, indeed has already been done.
Remark: Even though integration by parts works directly, the preliminary substitution $t=x^2$, as in the solution by Pranav Arora, is a better way to proceed.
Use the substitution $x^2=t \Rightarrow 2x\,dx=dt$ to get: $$\int x^3e^{x^2}\,dx=\frac{1}{2}\int te^t \,dt$$ The last integral can be evaluated using integration by parts: $$\frac{1}{2}\int te^t \,dt=\frac{1}{2}\left(te^t-e^t\right)+C=\frac{1}{2}e^t(t-1)+C$$ Since $t=x^2$, $$\frac{1}{2}e^t(t-1)+C=\boxed{\dfrac{1}{2}e^{x^2}(x^2-1)+C}$$