Integrate x^-1: Solutions & Explanation

• no name
In summary, integration x^-1 is one of the ways to define the natural logarithm of x, using a definite integral. It can also be shown that the derivative of ln(x) is equal to 1/x. This integration can be useful in computing the antiderivative of 1/x.
no name
find
integration x^-1
i tried but reached nothing ...

It's one of the ways to define ln(x). (Well, you use a definite integral for that, but it's practically the same).

$$\int_1^x{\frac{1}{t}dt}=\ln \left|x\right|$$

You have to be kidding,right...?

$$\int \frac{dx}{x}$$

,or is it a joke...?

And what's this doing in the "Brain Teaser forum"...?April 1-st was a while ago...

Daniel.

Dex you can be really mean sometimes.

I have a mean looking avatar,don't I ? Why should i be gentle then ?

I appologize to the OP,whose name i don't know and couldn't possibly find out...

Daniel.

Hehe, is that piccolo?

No,he's a waiter...

Can the OP compute the derivative of $\ln x$ ...?It would be useful to compute the antiderivative...

Daniel.

whozum said:
Hehe, is that piccolo?
No man, can't you see? That's Kami! Ofcourse he merged with Piccolo so now they are actually one. Good old Kami-sama was always pretty gentle actually. It was Piccolo who was mean.

could u prove it please ?

It can be shown starting with the definition that

$$\frac{d\ln x}{dx}=\frac{1}{x}$$.

Then

$$\int \frac{dx}{x}$$

becomes after the substitution

$$\frac{1}{x}=e^{t} \Rightarrow dx=-e^{-t} dt$$

$$\int e^{t}\left(-e^{-t} \ dt\right) =-\int dt=-t+C$$

Inverting the substitution,one finds

$$\int \frac{dx}{x}=-\ln\left(\frac{1}{\left|x\right|}\right) +C =\ln\left|x\right| +C$$

Daniel.

1. What is the definition of integration?

Integration is a mathematical process that involves finding the area under a curve. It is the inverse operation of differentiation and is used to find the original function from its derivative.

2. What is an x^-1 term in an integral?

An x^-1 term in an integral represents the inverse of x, or 1/x. This is also known as the reciprocal of x.

3. How do you solve an integral with an x^-1 term?

To solve an integral with an x^-1 term, you can use the power rule of integration. This involves increasing the power of x by 1 and dividing the result by the new power. In the case of x^-1, the result would be ln|x| + C, where C is the constant of integration.

4. What is the significance of the constant of integration in an integral?

The constant of integration represents any unknown constant in the original function. Since differentiation removes all constants, the constant of integration is added in the integral to account for these constants.

5. Can an integral with an x^-1 term have multiple solutions?

Yes, an integral with an x^-1 term can have multiple solutions. This is because the power rule of integration only gives one possible solution, but there could be other functions that have the same derivative. These functions would differ by a constant, which is accounted for by the constant of integration.

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