Question

What happens to the "3" in this integral?
∫3(3x - 1)^4 dx =∫(3x-1)^4(3)dx = (3x-1)^5/5 + C

What happens to the "2x+1" in this integral?
∫(2x + 1)(x^2 +x)dx = ∫(x^2+x)^1(2x+1)dx = (x^2+x)^2/2+ C

Answer 1

hey, your avatar is so cute

Answer 2

lol what's up with you guys and the penguins? XD

Answer 3

I am looking at his, it 's soooo cute.

Answer 4

lol you're isn't too shabby ;)

Answer 5

let zepdrix help you, cute penguin

Answer 6

\[\large\bf \frac{d}{dx}\frac{1}{5}(3x-1)^5+C\quad=\quad \frac{1}{5}\cdot5(3x-1)^4\color{orangered}{(3)}\quad=\quad \color{orangered}{3}(3x-1)^4\]Understand where the orange 3 is coming from? *cough cough* chan rule.

Answer 7

To really get a grasp on where the 3 is going, it's important that you understand how to apply a substitution.

Are you familiar with `u-substitution` or something similar? :)

Answer 8

Yeah, so in this problem, u^4 = (3x-1)^4 and du = 3dx

Answer 9

\[\Large \int\limits 3(3x - 1)^4 dx\quad=\quad \int\limits (\color{royalblue}{3x-1})^4(\color{orangered}{3dx})\]\[\Large \color{royalblue}{u=3x-1},\qquad\qquad \color{orangered}{du=3dx}\]Yes very good sinoa!

Answer 10

\[\Large =\quad \int\limits\limits (\color{royalblue}{u})^4(\color{orangered}{du})\]

Answer 11

The idea is, we replaced 3x-1 with something else,
and then we replaced the derivative of 3x-1 with something else as well.

So that's why the 3 is disappearing.

Answer 12

So is the constant of integration (C) due to the constant 3, or are those unrelated?

Answer 13

Unrelated.
This particular integral is an `indefinite` integral.
We didn't define a start and end point for the integral so the answer we get should include the `entire family of solutions`.

I can give a nice simple example if that will help.

Answer 14

\[\Large (x^2+7)'\quad=\quad 2x\]\[\Large (x^2+2)'\quad=\quad 2x\]
So if I asked you for to evaluate this integral:\[\Large \int\limits 2x\;dx\quad=\quad?\]What should it give us?
2x+7 ??
2x+2 ?
2x+something else?

Answer 15

bahhh i meant to type
x^2+7 ??
x^2+2 ?

Answer 16

Okay, so if we had defined boundaries, we could specifically figure out the original constant, but because it's indefinite, we need to include the +C

Answer 17

Yes, good! :)
The + C gives us the family of solutions that could exist.

Answer 18

Well thank you very much! You really took the time to explain the concepts, which I really appreciated.

Answer 19

np!