A concept from calculus that has always bothered me

[ronaldor9]

When one writes $$\int p(t) \, dt$$ is the constant of integration implied? I have always thought that it wasn't necessary to write $$\int p(t) \, dt +k$$. However, in my diff. eq. book the constant is ussually written out, why is this so?

In addition the book also writes $$\int_{t_o}^t p(t) \, dt +k$$ isn't the constant here unnecessary since we now have the limits of integration included?

 

[snipez90]

http://en.wikipedia.org/wiki/Constant_of_integration

In my honors calculus course, we were always asked to find AN antiderivative of some function, so I always left the constant out. Most of the time, it's just a hassle.

 

[l'Hôpital]

It's usually written because it's definite.

Suppose

$$y ' = f(x, y(x)) \ , y(x_0) = y_0$$

It does seem logical to say that (assuming f is integratable)

$$\int y' dx = \int f(x,y(x)) dx \rightarrow y = \int f(x,y(x)) dx + k$$

but since we have initial conditions, we can do better.

$$\int_{x_0}^{x} y'(t) dt = \int_{x_0}^{x} f(t,y(t)) dt \rightarrow y - y_0 = \int_{x_0}^{x} f(t,y(t)) dt$$
Thus,
$$y = \int_{x_0}^{x} f(t,y(t)) dt + y_0$$

Better?

 

[ronaldor9]

Thanks l'Hopital
Why is the later form preferred over the first form?

 

[l'Hôpital]

Simply because it actually involves the initial conditions.

 

[ronaldor9]

There is one part in my book where it writes $$R'(y)=Q(x_0,y)$$
and then by integration $$R(y)=\int_{y_0}^y Q(x_0,y)\,dy$$.

Shouldn't theauthor here have included a constant at the end of the integral as you have written in you example?

 

[HallsofIvy]

There is one part in my book where it writes $$R'(y)=Q(x_0,y)$$
and then by integration $$R(y)=\int_{y_0}^y Q(x_0,y)\,dy$$.

Shouldn't theauthor here have included a constant at the end of the integral as you have written in you example?

Yes, because with that notation R(y₀)= 0 which is not true for all anti-derivatives. I would also object to using "y" both as a limit of integration (and so outside the integral) and as the variable of integration. Much better would be either
$$R(y)= \int_{y_0}^y Q(x_0,t)dt+ C$$
or
$$R(y)= \int^y Q(x_0, t)dt$$
where we don't need the "C" because the lower limit of integration is left open.

If you write just
$$R(y)= \int Q(x_0,y)dy$$
the usual notation for the "anti-derivative", the constant C is implied. You do not write it there. Of course, if you wrote
$$R(y)= \int y^2 dy= \frac{1}{3}y^2+ C$$
The constant on the right is necessary.