# Find the definite integral of a vector

## Homework Statement

If $\vec{r}(t) = t^2\vec{i} + t\cos{(\pi t)}\vec{j} + \sin{(\pi t)}\vec{k}$, evaluate $\int_{0}^{1} \vec{r}(t) \text{dt}$.

## The Attempt at a Solution

So I tried integrating each individual part, and I got
$\frac{1}{3}t^3\vec{i} + (-\pi t\sin{(\pi t)} - \frac{\sin{(\pi t)}}{\pi})\vec{j} - \frac{\cos{(\pi t)}}{t}\vec{k} |_{0}^{1}$
(For the coefficient of $\vec{j}$ I used integration by parts, I'm not sure if that's right because it looks weird)
But evaluating at 0 makes the coefficient of $\vec{k}$ undefined! What should I do? Thanks.

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SammyS
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## Homework Statement

If $\vec{r}(t) = t^2\vec{i} + t\cos{(\pi t)}\vec{j} + \sin{(\pi t)}\vec{k}$, evaluate $\int_{0}^{1} \vec{r}(t) \text{dt}$.

## The Attempt at a Solution

So I tried integrating each individual part, and I got
$\frac{1}{3}t^3\vec{i} + (-\pi t\sin{(\pi t)} - \frac{\sin{(\pi t)}}{\pi})\vec{j} - \frac{\cos{(\pi t)}}{t}\vec{k} |_{0}^{1}$
(For the coefficient of $\vec{j}$ I used integration by parts, I'm not sure if that's right because it looks weird)
But evaluating at 0 makes the coefficient of $\vec{k}$ undefined! What should I do? Thanks.
You do realize that
$\displaystyle \int_{0}^{1} \vec{r}(t)\,dt=\hat{i}\int_{0}^{1} t^2\,dt+\hat{j}\int_{0}^{1} t\cos(\pi t)\,dt- \hat{k}\int_{0}^{1} \sin(\pi t)\,dt\ ,$​
don't you?

Show how you did the integration by parts, and show how you get a t in the denominator of the k component.