Complex integral $\int_C{e^{z^3}z^2}dz$ over a parameterized curve $z(t)=t+t^{10}i$

Question

Compute $\int_C{e^{z^3}z^2}dz$ where C is given by $z(t)=t+t^{10}i$, where $t\in [0,1]$.

I think this is easier than I am making it, but can we just do u-substitution on this, and then integrate the $e^u \,du$, or is this more complicated than I am thinking? I have been overthinking these types of problems quite a lot and I'm unsure why.

Answer 1

Inasmuch as the antiderivative of $z^2e^{z^3}$ is $\frac13 e^{z^3}$, we have

$$\begin{align} \int_C z^2e^{z^3}\,dz&=\left.\left(\frac13e^{z^3}\right)\right|_{z=0}^{z=1+i}\\\\ &=\frac13 \left(e^{(1+i)^3}-1\right) \end{align}$$

And you can simplify.