Q. 5 - Calculus

Question

Calculus of vector-valued functions: Calculate each of the following.

$\int r\left(t\right)\mathrm{d}t, \mathrm{where} r\left(t\right)=e^t\mathbf{i}+t^3\mathbf{j}-4\mathbf{k}$

Step-by-Step Solution

Answer

The value of $\int r\left(t\right)\mathrm{d}t=e^t\mathbf{i}+\frac{t^4}{4}\mathbf{j}-4t\mathbf{k}$

Step 1.Given Information

Calculus of vector-valued functions: Calculate each of the following.

$\int r\left(t\right)\mathrm{d}t, \mathrm{where} r\left(t\right)=e^t\mathbf{i}+t^3\mathbf{j}-4\mathbf{k}$

Step 2. Now finding the value of ∫ r ( t ) d t

$$\begin{gathered} \int r\left(t\right)\mathrm{d}t=\int \left(e^t\mathbf{i}+t^3\mathbf{j}-4\mathbf{k}\right)\mathrm{d}t \\ \int r\left(t\right)\mathrm{d}t=\left(\int e^t\mathrm{d}t\right)\mathbf{i}+\left(\int t^3\mathrm{d}t\right)\mathbf{j}-4\left(\int \mathrm{d}t\right)\mathbf{k} \\ \int r\left(t\right)\mathrm{d}t=e^t\mathbf{i}+\left(\frac{t^4}{4}\right)\mathbf{j}-4\left(t\right)\mathbf{k} \\ \int r\left(t\right)\mathrm{d}t=e^t\mathbf{i}+\frac{t^4}{4}\mathbf{j}-4t\mathbf{k} \end{gathered}$$