Question

can someone help me with this pls?
integral of 1/x^5. What i do is
1)flip it = x^-5
2)do the integral = x^-4/-4
3)flip it back over = -4/x^4
the answer is -1/4x^4, i do not know how they got that..

Answer 1

Let me just write out what you seem to be doing.

Answer 2

\[
\int \frac{1}{x^5}dx = \int x^{-5} dx = \frac{x^{-4}}{-4} = \frac{-4}{x^4}
\]That last step doesn't make sense.

These two are already equal \[
\frac{1}{x^5} = x^{-5}
\]You don't ever flip them.

Answer 3

how did 4 end up on top?

Answer 4

But if you did call this "flipping" them, then you need to have \(-4\)'s power to change as well: \[
\frac{(-4)^{-1}}{x^4} = -\frac{1}{4x^4}
\]

Answer 5

When he "flipped" the fraction

Answer 6

\[x^{-n}=\frac{1}{x^{n}} \\ \text{ multiplying both sides by } K \text{ gives } K \cdot x^{-n}=K \cdot \frac{1}{x^{n}} \\ \\ K \cdot x^{-n} \neq \frac{1}{K} \frac{1}{x^n}\]

Answer 7

or if you choose to divide both sides by K
\[\frac{x^{-n}}{K}=\frac{1}{Kx^n} \\ \frac{x^{-n}}{K} \neq \frac{K}{x^n}\]
K shouldn't do anything magical

Answer 8

I'm not sure if someone's already cleared this up, but I remember not quite understanding this when it was first introduced to me. What you're doing isn't simply "flipping" a fraction. You're rewriting something in another way. For example, \[4^{-1} = \frac{1}{4}\]. So when you're flipping something, what you should really be doing is moving a single piece at a time. For example, \[4x^{-4}=\frac{4}{x^4}\] rather than "flipping" it and getting \[4x^{-4}=\frac{1}{4x^4}\]. Does this help?

Answer 9

Just simply apply the power rule: \(\dfrac{x^{n+1}}{n+1}\).