Question

Using an upper-case "C" for any arbitrary constants, find the general indefinite integral :
integral |3 3root(x) dx|

Answer 1

1. Take the 3 out of the integral. Always take constants out of the integral.
2. Assign u, which is x since it's derivative is 1dx, which is in the integral.
3. Take the antiderivative by adding 1 +1/3, which is 4/3. Take the 4/3 and put it in the denominator. The root of 3 = 1/3, which x is raised to.
4. The integral is gone, divide using keep change flip, and multiply across.
5. Your answer should be 9x^(4/3)/3 + C
If that wasn't clear, let me know.

Answer 2

The answer should have a denominator of 4, not 3.

Answer 3

whats the denominator?

Answer 4

Hmm this one might be a little bit trickier actually, due to the absolute bars.
I'm assuming that's what those are, yes?

Answer 5

zepdrix is right, is that an absolute value? I thought it was just to separate everything.

Answer 6

thats what the question looks like?

Answer 7

ok, so the denominator is also 4/3.

Answer 8

what is a denominator?

Answer 9

do i get the antiderivative to work out the ans?

Answer 10

Oh oh it's not an absolute :) That's a relief lol

Answer 11

how do i work it out if its an absolute?

Answer 12

When you have a fraction, there are two parts: the numerator and the denominator. The numerator is on top and the denominator is on bottom.
The antiderivative is the same as the integral. Yes, take the antiderivative to get the answer.

Answer 13

so the denominator is x? and the numerator is 3?

Answer 14

how would i go about getting the antiderivative of this?

Answer 15

\[\huge \int\limits3\; \sqrt[3]{x}\;dx\]This is what the problem looks like? :o

Answer 16

ya :(

Answer 17

I can't make the pictures so it's hard to explain, zeprdrix can so it's easier. After zepdrix helps you, there is a site called khanacademy.org that you can use. It's really good.

Answer 18

I'll just restate some of Fringes steps with the math tool, maybe it will help :)


Pull the constant outside of the integral, it won't affect the integration process.\[\large 3\int\limits \sqrt[3]{x}\;dx\]Rewrite the cube root, using a `rational expression`.\[\large 3\int\limits\limits x^{1/3}\;dx\]

From here, we just apply the `Power Rule for Integration`.
\[\large \color{royalblue}{\int\limits x^{n}dx\quad = \quad \frac{x^{n+1}}{n+1}}\]We increase the power by 1, then divide by the `new` power.

Answer 19

\[\large 3\left(\frac{x^{(1/3+1)}}{\dfrac{1}{3}+1}\right)\]

Answer 20

See how I put a +1 on the 1/3?
It's always a little messy looking with all these fractions :C

Answer 21

ya why did you put a +1?

Answer 22

Remember how to take a derivative?
It's 2 steps

~Multiply by the `old` power.
~Decrease the power by 1.

With integration we're doing the opposite, and in the opposite order.

~Raise the power by 1.
~Divide by the `new` power.

Answer 23

It might be easier to grasp the concept if you see an easy example,

Answer 24

we do the antiderivative yes?

Answer 25

so how would i go about finding the intergral with upper case C?

Answer 26

Anti derivative is `Integration`.
Yes :)

Answer 27

That has been explained to you several times now :) lol
Try to follow the steps posted above :O

Answer 28

i dont understand it

Answer 29

i think i figured it out!

Answer 30

Here's an example.\[\large (x^3)'\]If we wanted to take the derivative of this,
~We wring the 3 (the power) down as multiplication.
~Then we -1 the power.

\[\large 3x^{3-1}=3x^2\]______________________________________________

Now let's find the anti derivative of \(3x^2\).\[\large \int\limits 3x^2\;dx\]When we anti differentiate, the `big swirly S bar` and `dx` will disappear as part of this process.

~We'll +1 the power,
~Then we divide by the power.
\[\large \frac{3x^{2+1}}{2+1} \quad = \quad \frac{3x^3}{3} \quad = x^3\]

Answer 31

Seems like you're having some trouble with the idea of anti-derivatives. Just realize that it's the opposite process as taking a Derivative.

See how we got back what we started with in this example?