Question

what is the purpose of indefinate integrals?

Answer 1

The purpose is to find the family of anti-derivatives. There isn't any other notation for "find the anti-derivative".

Answer 2

ok, i understand that.

are like formulas for anti derivaties that I couldn't find on my own because they are too complex?

Answer 3

There are methods for finding it, and they come from reverse engineering the differentiation methods.

Answer 4

i'm probably not making a lot of sense.

what I am saying is when would I use the table of indefinate integrals? and why wouldn't I just solve the problem by hand?

Answer 5

Those are there in case you don't have access to a computer algebra system, and the particular integral would take a very long time by hand. There isn't always a clear method for finding solving an integral either.

Answer 6

ooohhh ok

so some integrals can be solved by hand using the basic ideas of integration and substitution, but when these are not able to do the job, we use indefinate integral tables, do I understand what you are saying?

Answer 7

First thing: Not all functions have an elementary anti-derivative. For example \(\sin(x^2)\) does not have an elementary anti-derivative. You would have to express its anti-derivative as an infinite sum.

Second thing: Some anti-derivatives can only be found by using subtle substitutions which are not at all obvious. There just is no reason you would normally think to use a certain substitution. Consider trig substitutions and how confusing they are.

So we aren't doing some simple calculation all the time. That is why they give you the tables.

Answer 8

awh ok ok, this has helped a lot, thanks

Answer 9

lets say I had the integral of sqrt(4y^2 + 1) evaluated from 0 to 1

Answer 10

I keep using substitution but I end up with

du* (1/8) = y*dx

and cant plug that back into the original, so I would use indefinate integrals?

Answer 11

Why is there a \(dx\)?

Answer 12

haha sorry, dy

\[\int\limits\limits_{0}^{1}\sqrt{4y ^{2}+1} dy\]

Answer 13

what substitution are you doing?

Answer 14

just u substitution

u = 4y^2 +1
du = 8y dy
du * (1/8) = y dy

Answer 15

ya?

Answer 16

True, but you don't have \(y\;dy\) in your integral.

Answer 17

ya, thats why I was asking to begin with what are the uses of indefinate integral tables

Answer 18

You could say: \[
\frac{u-1}{4}=y^2\implies y=\sqrt{\frac{u-1}{4}}
\]Since we know \(y\geq0\).

Answer 19

It leads you to another ugly integral.

Answer 20

...

Answer 21

In this case, I think the expectation is that you do a trig sub, but if you didn't know about that then you'd look it up in the table.

Answer 22

so, there are several methods to solving this based on what level in calculus i am at?

Answer 23

Yeah

Answer 24

For example, your \(u\) sub was still valid, it's just that it wasn't going to make the problem any simpler.

Answer 25

knowing which method to use depends on the situation and then from experience i'll get an 'eye' of what I need to do.

thanks for your help, textbooks can be so hard to understand sometimes