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if we apply ordinary integration formula to it we'll get [t^0 / 0] regardless of the limits of integration. can someone show me how to prove mathematically that this method is invalid to use and why
O.J. said:why did they exclude -1?
What do you mean by "prove mathematically". t^0/0= 1/0 is "undefined". If we set 1/0= x then we have 1= 0*x which is not true for any possible x.O.J. said:if we apply ordinary integration formula to it we'll get [t^0 / 0] regardless of the limits of integration. can someone show me how to prove mathematically that this method is invalid to use and why
O.J. said:if its undefined it mathematically means NO REAL number satisfies it, so why do we go on to find another nway of calculating this integral and why do we use logarithms for it?
O.J. said:wat i really mean is since 1/0 is UNDEFINED, can u formally assert that even tho its true 1/t still has an integral provided u have known limits?
HallsofIvy said:Many books define a "new function" by
[tex]log(x)= \int_0^x \frac{1}{t} dt[/tex]
and then prove that this is, in fact, the usual natural logarithm function.
Jarle said:What does the integral mean, does anyone have a link to a proper explanation of it's function?
Jarle said:What does the integral mean, does anyone have a link to a proper explanation of it's function?
In calculus, the integral of a function is an extension of the concept of a sum. The process of finding integrals is called integration. The process is usually used to find a measure of totality such as area, volume, mass, displacement, etc., when its distribution or rate of change with respect to some other quantity (position, time, etc.) is specified. There are several distinct definitions of integration, with different technical underpinnings. They are, however, compatible; any two different ways of integrating a function will give the same result when they are both defined.
The term "integral" may also refer to antiderivatives. Though they are closely related through the fundamental theorem of calculus, the two notions are conceptually distinct. When one wants to clarify this distinction, an antiderivative is referred to as an indefinite integral (a function), while the integrals discussed in this article are termed definite integrals.
The integral of a real-valued function f of one real variable x on the interval [a, b] is equal to the signed area bounded by the lines x = a, x = b, the x-axis, and the curve defined by the graph of f. This is formalized by the simplest definition of the integral, the Riemann definition, which provides a method for calculating this area using the concept of limit by dividing the area into successively thinner rectangular strips and taking the sum of their areas
O.J. said:the integral of 1/x to be a logarithm function whose base happens to be a number that 2.721...etc.
O.J. said:ur not gettin my point... let me put it this way. how did the founders of calculus explain their approach to calculating the int of 1/x. they must've provided an explanation along with it.
Adding the Exponents: If b is any positive real number then
bx by = bx+y
for all x and y. This is the single most important identity concerning logs and exponents. Since ex is only a special case of an exponential function, it is also true that
ex ey = ex+y
Multiplying the Exponents: If b is any positive real number then
(bx)y = bxy
for all x and y. Again since ex is a special case of an exponential function, it is also true that
(ex)y = exy
Converting to roots to exponents: The nth root of x is the same as
x1/n
for all positive x. Since square roots are a special case of nth roots, this means that
_
√x = x1/2
In addition:
__
√ex = ex/2
Converting to ex form: If b is any positive real number then
bx = ex ln(b)
for all x. This includes the case where you have xx:
xx = ex ln(x)
or if you have f(x)x:
f(x)x = ex ln(f(x))
or if you have xf(x):
xf(x) = ef(x) ln(x)
or if you have f(x)g(x):
f(x)g(x) = eg(x) ln(f(x))
As an example, suppose you had (x2 + 1)1/x. That would be the same as
e(1/x) ln(x2 + 1)
ex is its own derivative: The derivative of ex is ex. This is the property that makes ex special among all other exponential functions.
ex is always positive: You can put in any x, positive or negative, and ex will always be greater than zero. When x is positive, ex > 1. When x is negative, ex < 1. When x = 0 then ex = 1.
The log of the product is the sum of the logs: Let b, x, and y all be positive real numbers. Then
logb(xy) = logb(x) + logb(y)
This is the most important property of logs. Since ln(x) = loge(x), it is also true that
loge(xy) = ln(xy) = loge(x) + loge(y) = ln(x) + ln(y)
The log of the reciprocal is the negative of the log: For any positive b, x, and y
logb(1/x) = -logb(x)
logb(y/x) = logb(y) - logb(x)
This includes
ln(1/x) = -ln(x)
ln(y/x) = ln(y) - ln(x)
Concerning multiplying a log by something else: Let b and x be positive and k any real number. Then
k logb(x) = logb(xk)
This includes
k ln(x) = ln(xk)
It also means that
_
logb(√x) = (1/2)logb(x)
and
_
ln(√x) = (1/2)ln(x)
Converting log bases to natural log You can compute any base log using the natural log function (that is ln) alone. If b and x are both positive then
logb(x) =
ln(x)
ln(b)
Every log function is the inverse of some exponential function: If b is any positive real number, then
blogb(x) = logb(bx) = x
The right-hand part of this equation is true for all x. The left-hand part is true only for positive x. The functions, ex and ln(x) are also inverses of each other.
eln(x) = ln(ex) = x
The same rules for x apply as above.
The derivative of the natural log is the reciprocal: If x is positive, it is always true that the derivative of ln(x) is 1/x.
To find the derivative of logs of other bases, apply the conversion rule. So for the derivative of logb(x) you end up with
1
x ln(b)
The natural log can be expressed as a limit: For all positive x
xh - 1
ln(x) = lim
h -> 0 h
You can only take the log of positive numbers: If x is negative or zero, you CAN'T take the log of x -- not the natural log or the log of any base. In addition, the base of a log must also be positive. As x approaches zero from above, ln(x) tends to minus infinity. As x goes to positive infinity, so does ln(x). So ln(x) has no limit as x goes to infinity or as x goes to zero.
Natural log is positive or negative depending upon whether x is greater than or less than 1: If x > 1, then ln(x) > 0. If x < 1, then ln(x) < 0. If x = 1 then ln(x) = 0. Indeed the log to any base of 1 is always zero.
Something you Can't Do with Logs
There is no formula for the log of a sum: Don't go saying that log(a+b) is equal to log(a) log(b) because this is NOT TRUE.