Problem with Integrals

Question

I would like to solve that problem:

Let say I define $F[x]$ function:

F[x_] := Integrate[x^2, x]

Then I want to calculate eg. $F[2]$, but Mathematica emits an Integrate::ilim error ("Invalid integration variable or limit(s) in 2.") and throws me something strange: $\int 4 d2$. Why it puts "$d2$" instead of "$dx$"? How to solve it, so I can easily use my $F[x]$ function?

We all know that $\int x^2 dx = \frac{x^3}{3}$, so the function $F(x)$ should be equal $F(x)=\frac{x^3}{3}$ and then $F(2) = \frac{2^3}{3}=\frac{8}{3}$ but it's not :/

One way is f[y_] := Integrate[x^2, x] /. x :> y then f[2] yields 8/3. — corey979, Jan 04 '17 at 00:29
You should use Set instead of SetDelayed in this instance. More information can be found in Understand the difference between Set (or =) and SetDelayed (or :=). — Edmund, Jan 04 '17 at 00:30
Daniel, I'm just quite new to this, that's why I don't understand many things yet :P corey979, thanks, that works :) Edmund, thanks too, I'll read about this :) — SmokuZnadPotoku, Jan 04 '17 at 00:30
@corey979 That will work but is kind of slow since the integral is performed on each call. I prefer to use Set as the integral is performed once at assignment; f[x_] = Integrate[x^2, x]. — Edmund, Jan 04 '17 at 00:33
@corey979 Seems more complicated and memory intensive than just using Set. — Edmund, Jan 04 '17 at 00:35
@Edmund Agreed. Nevertheless, the OP might learn from searching the documentation what /. and :> are. — corey979, Jan 04 '17 at 00:37
You might want to pay attention to the error message. It's trying to tell what the problem is. Not that it's easy to understand, but if you learn to understand them, the messages are in fact helpful. E.g., why you do think Integrate is using 2 as a *variable* of integration in F[2]? — Michael E2, Jan 04 '17 at 00:40

m_goldberg · Accepted Answer · 2017-01-04T00:46:33.617

You want to do the integration first, then use the result of the integration to define the function. That is don like this:

Block[{x}, f[x_] = Integrate[x^2, x]];

Then

Definition[f]

f[x_] = x^3/3

and

f[2]

8/3

which is what you expect. The way you defined F, the integration expression is held unevaluated until F is called. When is F is called with 2, then 2 replaces each occurrence the symbol x in the integral expression and then the integral is evaluated. That is,

Integrate[x^2, x]

becomes

Integrate[2^2, 2]

You need the learn that difference between SetDelayed ( := ) and Set ( = ). The difference is both important and useful when defining functions in Mathematica.

Problem with Integrals

1 Answers1