Question

Find the indefinite integral. $$ \int\left(\ln t \mathbf{i}+\frac{1}{t} \mathbf{j}+\mathbf{k}\right) d t $$

Short answer

The integral of the given vector field is \((t \ln t - t + C) \mathbf{i} + (\ln | t | + C') \mathbf{j} + (t + C'')\mathbf{k}\).

Step-by-step solution

Deconstruct the Vector Field

Separate the vector field into its components: \(\ln t\), \(\frac{1}{t}\) and 1, and treat them as separate integrals: \(\int \ln t \, dt\), \(\int \frac{1}{t} \, dt\), and \(\int dt\) respectively.

Integrate Each Component

To find the indefinite integral of the first component, use the integral of the natural logarithm, which is \(t \ln t - t + C\). The integral of \(\frac{1}{t}\) is \(\ln | t | + C'\) and the integral of a constant 1 with respect to \(t\) is simply \(t + C''\). Remember to include the constant of integration \(C\), \(C'\), and \(C''\) in each integral.

Combine the Components

Combine the results of the three integrals back into one vector. This gives the resulting vector field: \((t \ln t - t + C) \mathbf{i} + (\ln | t | + C') \mathbf{j} + (t + C'')\mathbf{k}\).