Evaluating the Integral of ln(1+t)t^3/(1+t)

[utkarshakash]

Homework Statement

$\int \left( \dfrac{ln(1+\sqrt[6]{x})}{\sqrt[3]{x} + \sqrt{x}} \right) dx$

The Attempt at a Solution

Let x^(1/6) = t

$\int \dfrac{ln(1+t)t^3}{1+t} dt$

 

[Simon Bridge]

You want to evaluate:
$$\int \left( \dfrac{\ln(1+\sqrt[6]{x})}{\sqrt[3]{x} + \sqrt{x}} \right) dx$$

You attempted it by doing the substitution: $x^{(1/6)} = t$ - which gets you:

$$\int \dfrac{\ln(1+t)t^3}{1+t} dt$$
... and, from there, you get stuck?

Are you missing a factor of 6 in there?

Have you tried putting u=1+t ?

 

[utkarshakash]

You want to evaluate:
$$\int \left( \dfrac{\ln(1+\sqrt[6]{x})}{\sqrt[3]{x} + \sqrt{x}} \right) dx$$

You attempted it by doing the substitution: $x^{(1/6)} = t$ - which gets you:

$$\int \dfrac{\ln(1+t)t^3}{1+t} dt$$
... and, from there, you get stuck?

Are you missing a factor of 6 in there?

Have you tried putting u=1+t ?

I forgot that 6. By the way it won't make my life easier. I tried substituting z=1+t as well and came up with this.

$\displaystyle \int \dfrac{lnz(z-1)^3}{z} dz$

Also, how do you get your integral sign bigger? Mine renders as small when I use \int. Here I've used \displaystyle to get it bigger.

 

[Simon Bridge]

If you want to know how I get a particular format for something, use the "quote" button at the bottom of my posts - it shows you the markup.

Your int signs render small because you are using in-line math style - the "itex" or double-hash tags. To get things to work better, use the displaymath style - the "tex" or double-dollar tags.

Similarly, you can get standard functions to render properly by putting a backslash in front of them ... like \ln for the natural logarithm.

After the substitution you should have something of form $\int f(z)\ln|z|\; dz$ ...

Arm yourself with a bunch of tables to help your strategy:
http://en.wikipedia.org/wiki/List_of_integrals_of_logarithmic_functions
http://en.wikipedia.org/wiki/List_of_logarithmic_identities

 

[clamtrox]

Or you can use partial integration to massage the logarithm out (although things will get slightly messy)