Determine the antiderivative
capital 𝐹 of the function lower case 𝑓 of 𝑥 equals five 𝑥 to the fourth plus
four 𝑥 cubed where capital 𝐹 of one equals negative two.
Before we do anything else, we’ll
calculate the general antiderivative. And that means we’ll follow the
same process from the previous example. We’ll pull out the constant, add
one to our exponent, and then divide by the value of the new exponent. In this case, we’ll have five times
𝑥 to the fifth power divided by five. And we’ll reduce that to 𝑥 to the
fifth. Now, for the second term, take out
that four, we’ll raise 𝑥 cubed to 𝑥 to the fourth power, and then divide by
four. Which reduces to 𝑥 to the fourth
power. The four in the numerator and the
denominator cancel out.
If we were finding the general
form, we would add a constant 𝑐. And we say that capital 𝐹 of 𝑥
equals 𝑥 to the fifth power plus 𝑥 to the fourth power plus 𝑐. And we wanna plug in 𝐹 of one to
help us find the value of 𝑐. 𝐹 of one equals negative two. One to the fifth power plus one to
the fourth power. One plus one equals two. So two plus 𝑐 has to equal
negative two. Subtract two from both sides. And we see that the constant value
is negative four. We’ll take that information and
plug it in to what we found for the general antiderivative. An antiderivative under these
conditions is 𝑥 to the fifth plus 𝑥 to the fourth minus four.