Tricky Logarithmic Indefinite Integral

[Liquid7800]

Homework Statement

Hi,
Our professor has only taught us these methods for Integration...thus far:

• 
  Direct Integration

• 
  Substituion Method

So theoretically we should be able to solve this problem without using Integration by parts or partial fractions...:

Integrate X^(2x)*ln(x+1)

It should be solvable by substitution, direct integration...(double substitution) by algebraic manipulation etc.

Homework Equations

The Attempt at a Solution

This is my attempt at a solution...I am concerned I am performing illegal operations, and I would appreciate some help in correcting my errors...
thanks...(see attached)

 

[Dick]

There is some correct stuff in there. Did you notice that point at which ln(x)+1 turned into ln(x+1)? That's not legal. Things went downhill from there. I would suggest you write x^(2x) as e^(2x*ln(x)) and substitute u=x*ln(x). It might go easier. What's du? Uh, actually that may be the whole problem. I think ln x+1 is supposed to be ln(x)+1 not ln(x+1). They are very different.

 

[Liquid7800]

What's du? Uh, actually that may be the whole problem. I think ln x+1 is supposed to be ln(x)+1 not ln(x+1). They are very different.

Thanks...I think you are right...it probably is ln(x)+1 not ln(x+1) (no WONDER it seemed so weird)...I will try again and see what happens..

Thanks a lot!

 

[Liquid7800]

Hi,

Here is my new attempt at this problem...I follwed your suggestions and here is what I came up with...
I appreciate any feed back or suggestions. Thank you!

 

[Bohrok]

You can simplify your answer, but you also didn't integrate e^2u correctly.

I think it would have been just as easy, if not easier, to use u = x^2x. Then du turns out real nice.

 

[Liquid7800]

Thanks for the heads up...

You can simplify your answer, but you also didn't integrate e2u correctly.

Your right!...my bad.

I re-did the last part with another round of substitution and finally got:

e^2(x Ln(x))
2

or

X^2x
2

Does this seem right?

 

[Bohrok]

Thanks for the heads up...



Your right!...my bad.

I re-did the last part with another round of substitution and finally got:

e^2(x Ln(x))
2

or

X^2x
2

Does this seem right?

That's right. You could always check an integral by differentiating it.