How to Solve Integral of e^x ln(x)dx in Homework?

[madah12]

Homework Statement

find integral e^x ln(x)dx

Homework Equations

integral udv=uv-integral vdu

The Attempt at a Solution

u=ln(x) ,du=1/x dx
dv=e^xdx ,v=e^x


integral e^x ln(x)dx=ln(x)e^x -integral e^x/x dx

integral e^x/x dx =
u=e^x du=e^x dx , dv=1/x dx , v=ln(x)
ln(x)e^x-integral ln(x)e^xdx
then when I substituted I got 0=0 so what did I do wrong ?

 

[micromass]

If you get 0=0, that indicates that you did nothing wrong. In fact, everything you did is correct. It is just not the right approach to solving this problem.

In fact, the integral you're trying to calculate, can not be evaluated with elementary functions. That means that you can simply not give an easy answer to this question, nobody can...

 

[madah12]

But doesn't the fundamental theorem of calculus guarantees the existence of the anti derivative? so why can't we check the relation between the values of x and the values of the integral to come up with an approximate model for it?

 

[hunt_mat]

In you notation use v=e^{x} and u=ln x

 

[madah12]

sorry I don't understand you are saying.?

 

[micromass]

But doesn't the fundamental theorem of calculus guarantees the existence of the anti derivative? so why can't we check the relation between the values of x and the values of the integral to come up with an approximate model for it?

Yeah, sure. The fundamental theorem of calculus guarantees the existence of an anti-derivative. But that doesn't mean you can find the anti-derivative. That means that you cannot write the anti-derivative in terms of sin, cos, log and other well-known functions. So, no matter what you try, you will not be able to find the anti-derivative. But you do know that it exists...

 

[Mark44]

But doesn't the fundamental theorem of calculus guarantees the existence of the anti derivative? so why can't we check the relation between the values of x and the values of the integral to come up with an approximate model for it?

The FTC guarantees that an antiderivative exists, but it doesn't say how to get it.

For definite integrals there are a number of techniques for obtaining approximations, such as the trapezoid rule, Simpson's rule, Gaussian quadrature, etc.

 

[madah12]

The FTC guarantees that an antiderivative exists, but it doesn't say how to get it.

For definite integrals there are a number of techniques for obtaining approximations, such as the trapezoid rule, Simpson's rule, Gaussian quadrature, etc.

ok it guarantees that the anti derivative exists but does it guarantee that it exists as function that for every x we know some specific method to get a y if so then why can't we express it? if not then what does it mean that it exists.

 

[xeno_gear]

ok it guarantees that the anti derivative exists but does it guarantee that it exists as function that for every x we know some specific method to get a y if so then why can't we express it? if not then what does it mean that it exists.

We aren't guaranteed that we know any specific method to get a y. The FTC just guarantees that an antiderivative exists.

Existence means just that -- it exists, it's out there, but existence and construction are two different things. We're only guaranteed existence here but that doesn't tell us how to construct (i.e., how to find) an antiderivative.

For example, if we want to evaluate \int 2x\, dx the FTC tells us that an antiderivative exists, but doesn't tell us how to find it. However, we know how to find it because we know 2x is the derivative of x^2. So then the FTC tells us
\int 2x\, dx = x^2 + C

But note that the FTC didn't tell us *how* to get x^2, it only told us that there exists some function that is an antiderivative of 2x.

Likewise, the FTC says e^x \ln x has an antiderivative but it doesn't tell us *how* to find it. An antiderivative of e^x \ln x is some function whose derivative is e^x \ln x but such a function cannot be written down using our known elementary functions, as micromass pointed out earlier. Another example of a function whose antiderivative exists but can't be written down with elementary functions is e^{x^2}.

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Another example of an "existence theorem" is the fact that the equation ax^2 + bx + c = 0 has two distinct real solutions IF b^2 - 4ac > 0
This gives us specific conditions which guarantee the existence of two distinct real solutions, but it doesn't tell us HOW to find them.

A "construction theorem" (one that actually tells you how to find something) would be something like the quadratic formula.. If ax^2 + bx + c = 0 then x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}

 

[angelssecret]

you can use this definition for e^x

e^x=1+x+(x^2/2!)+...+ (x^n/n!)