# Indefinite Integral: Justification for Dropping Absolute Value Bars

• phymatter
In summary, the author is discussing the justification for dropping absolute value bars while doing an indefinite integral. They mention that in some regions of the number line, the absolute value bars can be dropped and in other regions, they need to be kept. However, the author also suggests verifying if the minus sign will make a difference in the local anti-derivative. The conversation also brings up the possibility of dropping the absolute value bars in some cases, but the context would need to be considered.
phymatter
In a book while doing an indefinite integral the author first wrote (sec2 x)1/2 = |sec x| , fine , then the author says the following :
"since we are doing an indefinite integral we can drop the absolute value bars" ,
now what is the justification for this ?

No such justification exists at all.

Code:
No such justification exists at all.
does this mean that the assumption is wrong?

Yes.

Suppose you have an indefinite integral of this sort.

First, you should see that in some regions of the number line, where sec(x)>0, you may drop the absolute value sign, and use sec(x) instead.

In other regions of the number line, you need to use |sec(x)|=-sec(x)

Now, it may well happen that that minus sign will not make the local anti-derivative different there than for the other region, but that is actually something you need to VERIFY, rather than to make it into a presupposition.

Understood?

Yes.

Suppose you have an indefinite integral of this sort.

First, you should see that in some regions of the number line, where sec(x)>0, you may drop the absolute value sign, and use sec(x) instead.

In other regions of the number line, you need to use |sec(x)|=-sec(x)

Now, it may well happen that that minus sign will not make the local anti-derivative different there than for the other region, but that is actually something you need to VERIFY, rather than to make it into a presupposition.

i got you that it is not correct to drop the absolute value bars arbitrarily , but rather take 2 different cases for |f(x)|>=0 and <0 , then if we get it same we can say we " drop the absolute value bars " ,
but in this case ($$\int|sec(x)|$$) i don't think that we can drop the absolute value bars in any case , or is there any such possible condition ??

One would have to see the context. Perhaps the task was to do the integral, so (as stated above) there are two cases, but to evaluate one of the cases is enough because of their similarity to each other.

## What is an indefinite integral?

An indefinite integral is a mathematical concept used in calculus to find the antiderivative of a function. It is a fundamental tool for solving problems involving rates of change and accumulation.

## What is the justification for dropping absolute value bars in an indefinite integral?

The justification for dropping absolute value bars in an indefinite integral is based on the fundamental theorem of calculus, which states that the derivative of the integral of a function is equal to the original function. Since the indefinite integral is the inverse operation of differentiation, the absolute value bars are not necessary as they would cancel out when taking the derivative.

## When can you drop absolute value bars in an indefinite integral?

You can drop absolute value bars in an indefinite integral when the integrand is a continuous function. This is because the fundamental theorem of calculus only applies to continuous functions.

## Can absolute value bars be dropped in definite integrals as well?

No, absolute value bars cannot be dropped in definite integrals as they represent the signed area under a curve. Dropping the absolute value bars would result in the loss of this information, making the definite integral incorrect.

## Are there any cases where absolute value bars cannot be dropped in an indefinite integral?

Yes, absolute value bars cannot be dropped in an indefinite integral when the integrand contains a singularity, or a point where the function is undefined. In these cases, the absolute value bars are necessary to properly evaluate the integral.

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