Answer
$2\mathbf{\hat{i}}-4\mathbf{\hat{j}}+32\mathbf{\hat{k}}$
Work Step by Step
1) Anti-differentiate and evaluate $\int_{0}^{2}
(t\mathbf{\hat{i}}-
t^3\mathbf{\hat{j}}+
3t^5\mathbf{\hat{k}})dt$ using power rule.
$$\left[\frac{1}{2}t^2\mathbf{\hat{i}}-
\frac{1}{4}t^4\mathbf{\hat{j}}+
\frac{1}{2}t^6\mathbf{\hat{k}}\right]_{0}^{2}\\
=\left[\frac{1}{2}(2)^2\mathbf{\hat{i}}-
\frac{1}{4}(2)^4\mathbf{\hat{j}}+
\frac{1}{2}(2)^6\mathbf{\hat{k}}\right]-
\left[\frac{1}{2}(0)^2\mathbf{\hat{i}}-
\frac{1}{4}(0)^4\mathbf{\hat{j}}+
\frac{1}{2}(0)^6\mathbf{\hat{k}}\right]\\
=2\mathbf{\hat{i}}-4\mathbf{\hat{j}}+32\mathbf{\hat{k}}
$$
