To evaluate the integral $\int \vec{r}(t) \, dt$, where

$\vec{r}(t) = (t^2 - 5)e^{t^3 - 15t} \, \vec{i} + 4t\sqrt{t^2 + 1} \, \vec{j} - \sin^2(5t) \, \vec{k},$

we need to integrate each component of $\vec{r}(t)$ separately with respect to $t$:

1. $\vec{i}$-component: $\int (t^2 - 5)e^{t^3 - 15t} \, dt$
2. $\vec{j}$-component: $\int 4t\sqrt{t^2 + 1} \, dt$
3. $\vec{k}$-component: $\int -\sin^2(5t) \, dt$

Each of these integrals may require specific techniques:

• The $\vec{i}$-component involves an exponential function with a polynomial argument, potentially requiring substitution.
• The $\vec{j}$-component includes a square root term that might be simplified with a trigonometric or hyperbolic substitution.
• The $\vec{k}$-component is a squared sine function, which can be simplified using trigonometric identities, such as $\sin^2(5t) = \frac{1 - \cos(10t)}{2}$.

Would you like a detailed solution for each component?

1. How would substitution help in the $\vec{i}$-component's integration?
2. What trigonometric identity simplifies $\sin^2(5t)$?
3. Could a numerical approach be more efficient here?
4. How does integration affect the vector direction?
5. Can we use hyperbolic functions for the $\vec{j}$-component?

Tip: Breaking down complex integrals component-by-component often clarifies the steps needed for each part.