### Video Transcript

Calculate the indefinite integral
of four π‘ cubed plus three π‘ squared π’ plus four π‘ squared minus five π£ plus
four π‘ cubed minus five π‘ squared plus three π€ with respect to π‘.

This is a vector-valued
function. This takes a real number π‘. And it outputs a position
vector. And to integrate a vector-valued
function, we simply integrate each component in the usual way. So weβll integrate the component
for π’ with respect to π‘. Thatβs four π‘ cubed plus three π‘
squared. Weβll integrate the component for
π£. Thatβs four π‘ squared minus
five. And weβll integrate the component
for π€, four π‘ cubed minus five π‘ squared plus three with respect to π‘. These are polynomial functions. And we know that, to integrate a
polynomial term whose exponent is not equal to negative one, we increase that
exponent by one and then divide by that number. This means the integral of four π‘
cubed is four π‘ to the fourth power over four. Integrating three π‘ squared and we
obtain three π‘ cubed over three. And of course, this is an
indefinite integral. So we must have that constant of
integration π. Simplifying fully and we obtain π‘
to the fourth power plus π‘ cubed plus π.

Similarly, when we integrate four
π‘ squared, we get four π‘ cubed over three. The integral of negative five is
negative five π‘. And then, we need another constant
of integration π. In our final component, when we
integrate four π‘ cubed, once again, we get four π‘ to the fourth power over
four. The integral of negative five π‘
squared is negative five π‘ cubed over three. And the integral of three is three
π‘. Letβs have the final constant of
integration π. This simplifies as shown.

We put this back into vector
form. And we see that our integral is
equal to π‘ to the fourth power plus π‘ cubed plus π π’ plus four-thirds π‘ cubed
minus five π‘ plus π π£ plus π‘ to the fourth power minus five-thirds π‘ cubed plus
three π‘ plus π π€. Notice, though, that each of our
components has its own constant. So we can combine these and form a
constant vector π. And this means our integral is
equal to π‘ to the fourth power plus π‘ cubed π’ plus four-thirds π‘ cubed minus
five π‘ π£ plus π‘ to the fourth power minus five-thirds π‘ cubed plus three π‘ π€
plus this capital π, which represents a constant vector.