Understanding Ambiguous Notation in Math

[tony873004]

Homework Statement

$$\int {x^2 \sin \pi x\,dx}$$

What does this mean?
This:
$$\int {x^2 \sin \left( \pi \right)x\,dx}$$, in which case why didn't they write $$\int {x^3 \sin \left( \pi \right)\,dx}$$

Or this:
$$\int {x^2 \sin \left( {\pi x} \right)\,dx}$$, which I'm guessing is right, except that in a previous chapter I interpreted $$\sec \theta \,\tan \theta$$ to be $$\left( {\sec \theta } \right)\,\left( {\tan \theta } \right)$$

instead of
$$\sec \left( {\theta \,\tan \theta } \right)$$

Homework Equations

The Attempt at a Solution

 

[Dick]

I agree with both of your interpretations. Meaning sec(theta*tan(theta)) without parenthesizing is bizarre.

 

[Schrodinger's Dog]

In my experience books and particularly textbooks for study often play around with similar equations and present them in different ways, to get you used to the idea that they are indeed the same, and there are many ways of presenting equations.

It's kind of analogous to:-

$$\dot{x}(t)$$

$$\frac{dy}{dx}$$

$$\frac{d^2y}{dx^2}$$

$$f'(x)$$

$$f''(x)$$

They like to insert different forms of the same thing to make you think about equivalence and other ways of expressing the form of equations.

Or this:
$$\int {x^2 \sin \left( {\pi x} \right)\,dx}$$, which I'm guessing is right, except that in a previous chapter I interpreted $$\sec \theta \,\tan \theta$$ to be $$\left( {\sec \theta } \right)\,\left( {\tan \theta } \right)$$

instead of
$$\sec \left( {\theta \,\tan \theta } \right)$$

they are not the same as far as I can see. I guess if your going to interpret something make it equivalent in any form, if it isn't then you've made an interpretational error.


$$\sec \left( {\theta \,\tan \theta } \right)$$

$$=\sec\theta\sec\tan\theta}$$

not

$$\left( {\sec \theta } \right)\,\left( {\tan \theta } \right)$$

 

[ChaoticLlama]

I'm fairly sure your book means $$\int {x^2 \sin \left( {\pi x} \right)\,dx}$$, otherwise you would be integrating (0)x³!

 

[Schrodinger's Dog]

I'm fairly sure your book means $$\int {x^2 \sin \left( {\pi x} \right)\,dx}$$, otherwise you would be integrating (0)x³!

Yeah I notice a lot of people don't put say log(x) they put log x or say x^2.sin x instead using the notation x^2.sin(x) ? It's equivalent though. I'm sure that's what they mean. Looking around this forum alone shows a lot of variation between notation methods.

 

[cristo]

$$\sec \left( {\theta \,\tan \theta } \right)$$

$$=\sec\theta\sec\tan\theta}$$

This isn't true. Take $\theta=\frac{\pi}{4}$, then $$\sec (\theta\tan\theta)=\sec(\frac{\pi}{4}\cdot 1)=\sec(\frac{\pi}{4}). \hspace{1cm} \sec\theta\sec(\tan\theta)=\sec\frac{\pi}{4}\sec(1)$$. Since sec(1)≠1, the identity cannot hold for all theta.

 

[Schrodinger's Dog]

This isn't true. Take $\theta=\frac{\pi}{4}$, then $$\sec (\theta\tan\theta)=\sec(\frac{\pi}{4}\cdot 1)=\sec(\frac{\pi}{4}). \hspace{1cm} \sec\theta\sec(\tan\theta)=\sec\frac{\pi}{4}\sec(1)$$. Since sec(1)≠1, the identity cannot hold for all theta.

That's true, I was thinking of something else. same with sin(xcos(x)) my bad. Anyway, I think the intepretation that was given above is likely correct, my idiotic explanation aside :/ I'd go with post no 4.

 

[cristo]

In my experience, we include brackets when the expression we are writing is written in an unconventional way. For example, no one would look twice at sinx being anything other than sin(x). In the same way, I would write $\sin\pi x$without brackets, since if we were wanting to say sin(pi)x, we would write $x\sin\pi$. So, as the OP says, if it were meant to say $x^2\sin(\pi)x$, then we would write this as $x^3\sin\pi$

In a similar way, [tex]\sec\theta\tan\theta[/itex] is taken to mean $(\sec\theta)(\tan\theta)$.

 

[tony873004]

Thanks for all the replies

I'm fairly sure your book means $$\int {x^2 \sin \left( {\pi x} \right)\,dx}$$, otherwise you would be integrating (0)x³!

I figured this out when I tried it both ways.