Question

Find the indefinite integral. $$ \int\left(4 t^{3} \mathbf{i}+6 t \mathbf{j}-4 \sqrt{t} \mathbf{k}\right) d t $$

Short answer

The indefinite integral \(\int\left(4 t^{3} \mathbf{i}+6 t \mathbf{j}-4 \sqrt{t} \mathbf{k}\right) d t = t^{4}\mathbf{i} + 3t^{2}\mathbf{j} - \frac{8}{3}t^{3/2}\mathbf{k} + C\)

Step-by-step solution

Analyzing The Integrals

In the problem, recognize that each component is a separate function of \(t\). The vector \(4t^{3}\mathbf{i} + 6t\mathbf{j} - 4\sqrt{t}\mathbf{k}\) can be decomposed into three independent integrals.

Computing Component-Wise Integrals

Evaluate each integral independently. For the \(\mathbf{i}\)-component, \( \int 4t^{3} dt = t^{4}\); for the \(\mathbf{j}\)-component, \(\int 6t dt = 3t^{2}\); and for the \(\mathbf{k}\)-component, \( \int -4\sqrt{t} dt = -\frac{8}{3}t^{3/2}\). Remember to add the constant of integration \(C\) to each term.

Compiling The Results

Bring the results back together. The solution of the given integral is the vector obtained by collecting the results from the second step: \(t^{4}\mathbf{i} + 3t^{2}\mathbf{j} - \frac{8}{3}t^{3/2}\mathbf{k} + C\), where \(C\) may also be a vector, since we are dealing with vector functions.