Question

Find the indefinite integral. $$ \int\left(e^{t} \mathbf{i}+\sin t \mathbf{j}+\cos t \mathbf{k}\right) d t $$

Short answer

The indefinite integral of the given vector field is \( e^t \mathbf{i} - \cos(t) \mathbf{j} + \sin(t) \mathbf{k} \).

Step-by-step solution

Integration of the first term

Integrate \( e^t \) w.r.t \( t \). The integral of \( e^t \) is the familiar function itself. So, the integral of \( e^t \) is \( e^t \). Thus, the first term in the vector field, upon integration, becomes \( e^t \mathbf{i} \).

Integration of the second term

Next, integrate \( \sin(t) \) w.r.t \( t \). The integral of \( \sin(t) \) is \( -\cos(t) \). Hence, the second term in the vector field, upon integration, becomes \( -\cos(t) \mathbf{j} \).

Integration of the third term

Finally, integrate \( \cos(t) \) w.r.t \( t \). The integral of \( \cos(t) \) is \( \sin(t) \). So, the third term in the vector field, upon integration, becomes \( \sin(t) \mathbf{k} \).

Combine all Integrals

Combine all the integrated terms to construct the integral of the entire vector field. So, the indefinite integral of the vector becomes \( e^t \mathbf{i} - \cos(t) \mathbf{j} + \sin(t) \mathbf{k} \).