Question

In Exercises \(43-48,\) find the indefinite integral. $$ \int(2 t \mathbf{i}+\mathbf{j}+\mathbf{k}) d t $$

Short answer

The indefinite integral of \(2t\mathbf{i} + \mathbf{j} + \mathbf{k}\) is \((t^2 + C_1)\mathbf{i} + (t + C_2)\mathbf{j} + (t + C_3)\mathbf{k}\).

Step-by-step solution

Separate the components

First, separate the integral into three separate integrals each corresponding to a component i, j, k of the vector. So we have to find:\[\int 2t dt\mathbf{i}, \int dt\mathbf{j}, \int dt\mathbf{k}\]

Compute the integral for each component

Next, perform the integral on each component:For the \(\mathbf{i}\) component: \[\int 2t dt = t^2 + C_1\mathbf{i}\]For the \(\mathbf{j}\) component: \[\int dt = t + C_2\mathbf{j}\]For the \(\mathbf{k}\) component: \[\int dt = t + C_3\mathbf{k}\]Here, \(C_1, C_2, C_3\) are the constants of integration.

Combine the results

Finally, combine the results from each integral to get the final result:\[(t^2 + C_1)\mathbf{i} + (t + C_2)\mathbf{j} + (t + C_3)\mathbf{k}\]