Question

Proof with Integrals? Are you Up for this Challenge?

Answer 1

Prove that\[\int\limits_{}^{} f(x)dx= xf(x) - \int\limits_{}^{}xf'(x)dx.\]

Answer 2

integration by parts

Answer 3

Oh ok I got it

Answer 4

But there is another part to this problem...

Answer 5

That I am stuck on..

Answer 6

Let f and g be inverse functions, and assume f' is continuous. Prove that,

Answer 7

\[\int\limits_{a}^{b} f(x)dx= b* f(b) - a*f(a) - \int\limits_{f(a)}^{f(b)} g(y) dy\]

Answer 8

I would really appreciate help :)

Answer 9

I'm looking at one of the hints and it states to substitute y=f(x) in the second integral?

Answer 10

Oh and it also says to use part a and rewrite x using inverses

Answer 11

If thats the case, what would \(g(f(x)) = ? \) if we know g is an inverse. And what would \( d(f(x)) \) be?

Answer 12

I'm not sure to be honest.... :(

Answer 13

would it equal x?

Answer 14

Correct, \(g(f(x)) = x\). What would \(d(f(x))\) become, got an idea?

Answer 15

If you mean the derivative of f(x), it would become f'(x)?

Answer 16

well, remember the chain rule. In this case its the differential, but yes.

Answer 17

Wait just to clarify, if f and g are inverse functions g(f(x))=x or f(g(x))=x is always true?

Answer 18

correct

Answer 19

Okay, how do we continue?

Answer 20

Well, \( d(f(x)) = f'(x) dx \)

Answer 21

Can we use that and the first integral you posted?

Answer 22

No, since it is f'(x)...

Answer 23

Isn't that what is in the second integral of this equation:
\[\int f(x) dx = xf(x) -\int xf'(x) dx \]

Answer 24

IE: We can convert the integral in the inverse problem into the form we see above.

Answer 25

ooohh, I was looking at the wrong integral :)

Answer 26

no problem

Answer 27

just substitute f'(x)dx with the d f(x)?

Answer 28

yup, and the fact that \(g(f(x)) = x \)

Answer 29

could we set up f(x) as y and since the functions are inverses x= g(y) and so dy= f'(x)dx?

Answer 30

Notice too, when our domain of integration was \(dy\), the limits had to be \(f(a), f(b)\). Since the domain changes to \(dx\), we can use \(a,b\)

Answer 31

Yes, thats one way to look at it \(y = f(x) \) then \(dy = f'(x) dx\)

Answer 32

and it becomes a substitution problem, remember to change the limits too.

Answer 33

yes, like you mentioned... since the limits are x=a to x=b, we must integrate them into y=f(a) to y=f(b)

Answer 34

Or the other way around, since we are changing from \(y\) to \(x\)

Answer 35

oh okay

Answer 36

Okay, I'll try writing a solution for this problem.. Will you be willing to check it off please? :)

Answer 37

Sure, i'll try :) Will be sleeping soon

Answer 38

Since f and g are inverse functions, g(f(x))=x. Thus the derivative of f(x) is f'(x)dx. And so we convert the given equation into ∫f(x)dx=xf(x)−∫xf′(x)dx .

Answer 39

Oops, the question marks is the equation that you rewrote!

Answer 40

The text sounds right. And obviously the limits of evaluation arise from the fundamental theorem.

Answer 41

okok :) Thank you