Respuesta :

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[tex]\mathbf r(t)=\displaystyle\int(5t^4\,\mathbf i+6t^5\,\mathbf j+t\,\mathbf k)\,\mathrm dt[/tex]
[tex]\mathbf r(t)=(t^5+C_1)\,\mathbf i+(t^6+C_2)\,\mathbf j+\left(\dfrac12t^2+C_3\right)\,\mathbf k[/tex]

Given [tex]\mathbf r(1)=\mathbf i+\mathbf j[/tex], we have

[tex]\begin{cases}1^5+C_1=1\\\\1^6+C_2=1\\\\\dfrac121^2+C_3=0\end{cases}\implies C_1=0,C_2=0,C_3=-\dfrac12[/tex]

so that the particular solution is

[tex]\mathbf r(t)=t^5\,\mathbf i+t^6\,\mathbf j+\dfrac12(t^2-1)\,\mathbf k[/tex]

The function r(t) is found as, r(t) =t⁵i+6t⁵j+tk. Functions are required for the formulation of physical connections.

What exactly is a function?

A function is a statement, rule, or law that specifies the connection between two variables.

The given function is;

[tex]\rm r(t)= \int\limits^a_b ({5t^4i+6t^3j+tk} \, )dt[/tex]

For, r(1)=i+j

Substitute the value;

[tex]1^5+C_1=1\\\\ C_1=0 \\\\ 1^+C_2=1\\\\C_2=0\\\\ \frac{1}{2}\times 1^2 +C_3= 0\\\\ C_3=\frac{-1}{2}[/tex]

Hence,the function r(t) is found as, r(t) =t⁵i+6t⁵j+tk

To learn more about the function refer to:

https://brainly.com/question/5245372

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