Question

Does anybody know a good website that goes through a step to step process about how to integrate ?

Answer 1

I can help...and i'll give you examples to solve as well

Answer 2

thank you but i meant like lectures, i have an exam coming up and i'm really confused about the whole unit

Answer 3

it's part of first year university but my grade 12 teacher is making us do it now

Answer 4

Is it just basic integration? So i'm guessing that's basic integration.

Answer 5

it's basic, partial, area, and definite

Answer 6

I'm not sure what you mean by partial because integration by parts is more than basic integration. You would have to know exponential functions and log functions as well as natural functions.

Answer 7

natural log functions* and trig functions

Answer 8

But anyway, I can explain integration step by step with examples then if it doesn't help, i'll give you a link

Answer 9

okay thanks ! :)

Answer 10

i have a link to my teacher's website if you want to check out what he's doing

Answer 11

it's just not clear to me in words what he's doing and what method he applies

Answer 12

ok sure, what's the link?

Answer 13

http://sh.triton.net/math.htm the last calender for mhf4u the last few notes

Answer 14

ok so I assume you have done integration of log/natural log functions, exponential functions and trig functions?

Answer 15

yes

Answer 16

Ok so I opened the notes for May 29, what exactly don't you understand?

Answer 17

okay i understand everything up till the dy/dx
that really confuses me

Answer 18

like i just dont understand what to do when i see that

Answer 19

I don't think we have a dy/dx anywhere in integration by parts unless you are referring to du/dx? after substituting by u?

Answer 20

yes typo :$ !

Answer 21

so i just substitute it from there on or what?

Answer 22

ok I'll start from basics. In Integration we can have two functions multiplied together, this makes things complicated. So we either use u substitution or integration by parts.

Answer 23

okay

Answer 24

U substitution is usually much simpler. Let me give you a couple of examples.

Answer 25

okay that would be great

Answer 26

Evaluate \[\int\limits_{}^{} (x^2 +1)/(x^3+3x) dx\]

Answer 27

lett (x^3 + 3x) be u ?

Answer 28

If you want to substitute first look at what you are trying to substitute and see if it will give you the other function after differentiation.
In this case, u= x^3 + 3x because du=3(x^2+1)dx
notice you're du almost looks like x^2 +1 which is the other function that we are trying to integrate?

Answer 29

oh so the other function would be u ?

Answer 30

\[u= x^3 +3x\]
so once we have u, we differentiate it with respect to x. so that means
\[du/dx= 3x^2 +1\]
The reason why we do all this, is because we need to get rid of the dx, along with the other function, and just have one function that we can easily integrate.

Answer 31

oh okay !

Answer 32

I mean \[du/dx= 3x^2 + 3\]

Answer 33

okay

Answer 34

So we multiply both sides of the equation to get du= 3(x^2 + 1) dx

Answer 35

okay

Answer 36

do multipy

Answer 37

But then since I want to get rid of \[(x^2 +1) dx\] I have to make du equal to \[(x^2 +1) dx\]
This is because when we substituted x^3 + 3x with u, the integral looked like this
\[\int\limits_{}^{} (x^3+1) dx/u\]

Answer 38

why do we have to put the dx in ?

Answer 39

Into the integral you mean?

Answer 40

yes

Answer 41

Whenever you write an integral you will notice a dx dt dv, depending on what the function is. The dx,dt,dv or whatever it may be is part of the function that you are trying to integrate. So you can not ignore it.

Answer 42

okay thanks

Answer 43

So back to the example

Answer 44

yup

Answer 45

I have to make my du similar to (x^3 +1) dx so that I can replace the whole thing with du and be left with this integral\[\int\limits_{}^{} du/u\]

Answer 46

so is that the optimal ? like that's what format we want to reach ?

Answer 47

yes that is what you want to have, and then you can integrate from here.
The goal of substitution always, is to be left with a du and u. Could be
\[\int\limits_{}^{} udu\]
\[\int\limits_{}^{} du/u\]
\[\int\limits_{}^{} u^2 du\]
As long as we have a U and dU ONLY.

Answer 48

okay ill write this down so i remember :P

Answer 49

okay, so we had \[du= 3(x^2 +1)dx\] right?

Answer 50

yupp

Answer 51

So we divide both sides by 3 to get \[1/3 du= (x^2 + 1)dx\]

Answer 52

okayy

Answer 53

Now our du is exactly similar to the (x^2 + 1)dx in our integral, so we substitute this with 1/3 du to have \[\int\limits_{}^{} du/3u\]
because it's 1/3 du divided by u

Answer 54

ohh okay

Answer 55

but then since 1/3 is a constant we can put it before the integral to have \[1/3 \int\limits_{}^{} du/u\]

Answer 56

Constants ALWAYS factor out.

Answer 57

oh okay do they equal zero

Answer 58

no, you will multiply your constant to the value you get from your integral.

Answer 59

ohh okay !

Answer 60

Yea, so now we're on the last step, and this is when we integrate, remember this, \[\int\limits_{}^{} du/u = \ln(u) +C \] ?

Answer 61

yes

Answer 62

So the answer to our integral would be
\[1/3 \int\limits_{}^{} du/u= 1/3 (\ln u)+ C \]
and then we just plug in our initial value of u, which was x^2 + 3x to get
\[\ln \left| (x^3 +3x \right| + C\] The absolute values just mean that ln x>0 so you may ignore them.

Answer 63

oh okaay !

Answer 64

so just as recap of what we did, we start by looking for the best function to substitute,
the function whose derivative will be similar to the other function.
Then we find du/dx.
Then we equal du to the derivative of the function * dx. du=xdx for example.
Then we replace the function xdx with du in the integral.
and integrate our u with respect to du.

Answer 65

How about you try one example of integration by u substitution?

Answer 66

honestly thank you soooooo much !

Answer 67

thanks so much but i have to get off now
goodnight:)

Answer 68

I mean I can give you the problem and solution and you can try it later?

Answer 69

that would be great ! :)

Answer 70

\[\int\limits_{}^{} 3x+6/(x^2 +4x -3) dx\]

Answer 71

tyy !

Answer 72

solution is \[(3/2) \ln \left| x^2 + 4x -3 \right| + C\]

Answer 73

and your welcome. Good luck. Unfortunate we didn't get to do integration by parts. But good luck.

Answer 74

thankyou :)