Video Transcript
Determine the indefinite
integral having negative four multiplied by the fifth root of 𝑥 to the power of
nine plus eight all multiplied by the fifth root of 𝑥 squared with respect to
𝑥.
Let’s start by writing this
roots as powered. We know that the 𝑛th root of
𝑥 is equal to 𝑥 to the power of one over 𝑛. Once we’ve written our roots as
powers, we can then combine them with the existing powers, using the fact that
𝑥 to the power of 𝑛 to the power of 𝑚 is equal to 𝑥 to the power of 𝑛
multiplied by 𝑚. Therefore, 𝑥 to the power of
nine to the power of one-fifth becomes 𝑥 to the power of nine-fifths. And 𝑥 squared to the power of
one-fifths becomes 𝑥 to the power of two-fifths. Now, we can expand the
brackets, using the fact that 𝑥 to the power of 𝑛 times 𝑥 to the power of 𝑚
is equal to 𝑥 to the power of 𝑛 plus 𝑚. So our integral becomes the
integral of negative four 𝑥 to the power of eleven-fifths plus eight multiplied
by 𝑥 to the power of two-fifths with respect to 𝑥.
Here, we can use the power rule
for integration which tells us that the indefinite integral of 𝑥 to the power
of 𝑛 with respect to 𝑥 is equal to 𝑥 to the power of 𝑛 plus one over 𝑛 plus
one plus 𝐶. We can apply this rule to our
integral term by term. For the first term, we have
negative four 𝑥 to the power of eleven-fifths. Therefore, 𝑛 is equal to
eleven-fifths. When we integrate this term, we
get negative four multiplied by 𝑥 to the power of 𝑛 plus one. And 𝑛 plus one is simply
sixteen-fifths. So it’s 𝑥 to the power of
sixteen-fifths. And then we need to divide by
𝑛 plus one. So that’s dividing by
sixteen-fifths.
For the second time, we have
eight multiplied by 𝑥 to the power of two-fifths. Therefore, 𝑛 is
two-fifths. So we add eight multiplied by
𝑥 to the power of 𝑛 plus one which is 𝑥 to the power of seven-fifths. And we then divide by
seven-fifths. And we mustn’t forget to add
our constant of integration 𝐶. Now, all that remains to do is
to simplify. And so we obtain a solution
that the indefinite integral of negative four multiplied by the fifth root of 𝑥
to the power of nine plus eight all multiplied by the fifth root of 𝑥 squared
with respect to 𝑥 is equal to negative five multiplied by 𝑥 to the power of
sixteen-fifths over four plus 40 multiplied by 𝑥 to the power of seven-fifths
over seven plus 𝐶.
