Question

Find the indefinite integral. $$ \int\left(\frac{1}{t} \mathbf{i}+\mathbf{j}-t^{3 / 2} \mathbf{k}\right) d t $$

Short answer

The integral is: \( ln|t|\mathbf{i} + t\mathbf{j} - \frac{2}{5}t^{5/2}\mathbf{k} + \mathbf{C} \)

Step-by-step solution

Separate the Vector Components

Split the integral into three separate integrals, each for the i, j and k components: \[\int \frac{1}{t} dt\mathbf{i} + \int dt\mathbf{j} - \int t^{3 / 2} dt\mathbf{k}\]

Integrate the i Component

For the i component, we integrate \( \frac{1}{t} dt \). The integral of \( \frac{1}{t} dt \) is \( ln|t| \). So for the i component, we have:\[ln|t|\mathbf{i}\]

Integrate the j Component

The j component, \( \int dt \), simply integrates to \( t \). So for the j component, we have:\[t\mathbf{j}\]

Integrate the k Component

For the k component, we integrate \( t^{3/2} dt \). The integral of \( t^{3/2} dt \) is \( \frac{2}{5}t^{5/2} \). So for the k component, we have:\[\frac{2}{5}t^{5/2}\mathbf{k}\]

Combine the Components

Finally combine all the components to get the final result. Don't forget to include the integration constant \( C \) in the final result:\[ln|t|\mathbf{i} + t\mathbf{j} - \frac{2}{5}t^{5/2}\mathbf{k} + \mathbf{C}\]