Definite Integrals/Notation

If we have

f(x) = x^2+2x+2

, I assume I can replace x with y and say

f(y) = y^2+2y+2

as x and y are dummy variables?

I was wondering if I can do the same when

F(x)

is represented as a definite integral.

For example, find the definite integral

\displaystyle F(x^2) \mathrm {\ that \ satisfies} \ F'(x) = e^{-x^2} \ and \ F(0) = 0.

So, I can say that by the Fundamental Theorem of Calculus,

\displaystyle F(x) = \int _0^xe^{-t^2}dt \ \mathrm {and \ so}\ F(x^2) = \int _0^{x^2}e^{-t^2}dt

?

Thanks!

Reply 1

DFranklin

18

This is another "that looks to be worded really strangely" post from you (in particular "Find the definite integral F(x^2)" looks odd) but I would agree with what you've written otherwise.

Reply 2

Takeover Season

OP

8

Original post by DFranklin

This is another "that looks to be worded really strangely" post from you (in particular "Find the definite integral F(x^2)" looks odd) but I would agree with what you've written otherwise.

Thank you. Yes, the original question was something along the lines:

\displaystyle \mathrm {\ Find \ the \ definite \ integral} \ F(x) \mathrm {\ that \ satisfies} \ F'(x) = e^{-x^2} \ and \ F(0) = 0.

I was just wondering that if they asked me to instead, find the definite integral F(g(x)) where g(x) was not just 'x' on its own, then how would it change my answer? I just wasn't sure if it was as simple as replacing the limit x with the g(x).

At first, I was thinking that if I had

F(x^2)

, then I'd find

F'(x^2) \ \mathrm {or} \ \frac{d}{dx} F(x^2)

but that wouldn't help because the information given is related to

F'(x)

!

(edited 4 years ago)

Reply 3

DFranklin

18

It would be a *very* odd thing to ask. It's at the level where I'm thinking "I feel it's incorrect to write this, but I can't actually think of a rule that says you can't, and I think I know what you mean, so I'll go with it even though it's horrible".

A lecturer / author / etc. should know better.

Reply 4

Takeover Season

OP

8

Original post by DFranklin

It would be a *very* odd thing to ask. It's at the level where I'm thinking "I feel it's incorrect to write this, but I can't actually think of a rule that says you can't, and I think I know what you mean, so I'll go with it even though it's horrible".

A lecturer / author / etc. should know better.

Thank you, I was just wondering for some reason.

Reply 5

Takeover Season

OP

8

Original post by DFranklin

It would be a *very* odd thing to ask. It's at the level where I'm thinking "I feel it's incorrect to write this, but I can't actually think of a rule that says you can't, and I think I know what you mean, so I'll go with it even though it's horrible".

A lecturer / author / etc. should know better.

I was just wondering that if I have a function f where f is a function of x and y. But, y is also a function of x e.g. y = x^2
So

f\left(x,\:y\right)

and

y\left(x\right)

, then

f\left(x,\:y(x)\right)

, so can I say that

\frac{\partial }{\partial x}\left(f\left(x,y\right)\right)

. But, when I calculate this, I can't treat y as a constant right because it is a function of x?

So, it'd be differentiating each term of y with respect to x?

Also, in this case if you replace y with x^2, then it is equivalent to saying:

\frac{\partial }{\partial x}\left(f\left(x,y\right)\right)=\:\frac{d}{dx}\left(f\left(x,y\right)\right)

?

(edited 4 years ago)

Definite Integrals/Notation

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