Question

Compute the indefinite integral of the following functions. $$\mathbf{r}(t)=\left\langle t^{4}-3 t, 2 t-1,10\right\rangle$$

Short answer

Question: Compute the indefinite integral of the vector function \(\mathbf{r}(t)=\left\langle t^{4}-3 t, 2 t-1,10\right\rangle\). Answer: The indefinite integral of the given vector function \(\mathbf{r}(t)\) is \(\mathbf{R}(t)=\left\langle \frac{t^5}{5} - \frac{3t^2}{2} + C_1, t^2 - t + C_2, 10t + C_3\right\rangle\), where \(C_1, C_2, C_3\) are constants of integration.

Step-by-step solution

Integrate the first component

First, let's find the integral of the first component, which is \(t^{4}-3 t\). To find the integral of this component, we need to find the antiderivative with respect to t: $$\int (t^{4} - 3t) dt$$

Integrate the second component

Now, let's find the integral of the second component, which is \(2t - 1\). To find the integral of this component, we need to find the antiderivative with respect to t: $$\int (2t - 1) dt$$

Integrate the third component

Finally, we need to find the integral of the third component, which is 10. To find the integral of this component, we need to find the antiderivative with respect to t: $$\int 10 dt$$

Combine the results

Now that we have found the indefinite integrals for each component, we can put the results together in the format of a vector function, giving us the indefinite integral of the given vector function.