Integral Substitutions and Mathematica

[zared619]

Hi all. My professor gave us some integrals that Mathematica can't do, and we have to teach Mathematica how to do them. I got the first two, but I'm stuck with the u substitutions for these six. I know that I am supposed to make an attempt at a solution, but I've tried several different u substitutions to no avail. Sorry for the formatting.
Any help is appreciated

3.. ∫(x sin x ln(x sin x))/(1-Sqrt[1-Sqrt[x sin x]]) (x cos x + sin x)\[DifferentialD]x

4. ∫(cos x -x sin x) Sqrt[x cos x] Sqrt[1+x^3 cos^3 x]\[DifferentialD]x

5. ∫((1+ln x) Sqrt[1+x ln x])/Sqrt[x ln x] \[DifferentialD]x

6. ∫(1-2/x^3)Sqrt[1-(x+1/x^2)^2]\[DifferentialD]x

7. ∫(1+ln x)Sqrt[1-x^2(ln x)^2]\[DifferentialD]x

8. ∫x^x Sqrt[x ln x](1+ln x)\[DifferentialD]x

Note: My professor said that #8 will include a function called Erfi[x]. I have no idea what that is.

Again, any help is appreciated.

 

[SteamKing]

You are looking for a version of the Error Function:

http://en.wikipedia.org/wiki/Error_function

 

[pasmith]

Hi all. My professor gave us some integrals that Mathematica can't do, and we have to teach Mathematica how to do them. I got the first two, but I'm stuck with the u substitutions for these six. I know that I am supposed to make an attempt at a solution, but I've tried several different u substitutions to no avail. Sorry for the formatting.
Any help is appreciated

3.. ∫(x sin x ln(x sin x))/(1-Sqrt[1-Sqrt[x sin x]]) (x cos x + sin x)\[DifferentialD]x

The combination $x \sin x$ occurs frequently here as an argument to other functions and, by fortune or design, the derivative of $x \sin x$ is $x \cos x + \sin x$.

This suggests $u = x \sin x$ as a first substitution, although further substitutions may be necessary.

Similar first substitutions suggest themselves for the others, although further substitutions might be necessary.

 

[Mandelbroth]

Hi all. My professor gave us some integrals that Mathematica can't do, and we have to teach Mathematica how to do them. I got the first two, but I'm stuck with the u substitutions for these six. I know that I am supposed to make an attempt at a solution, but I've tried several different u substitutions to no avail. Sorry for the formatting.
Any help is appreciated

3.. ∫(x sin x ln(x sin x))/(1-Sqrt[1-Sqrt[x sin x]]) (x cos x + sin x)\[DifferentialD]x

4. ∫(cos x -x sin x) Sqrt[x cos x] Sqrt[1+x^3 cos^3 x]\[DifferentialD]x

5. ∫((1+ln x) Sqrt[1+x ln x])/Sqrt[x ln x] \[DifferentialD]x

6. ∫(1-2/x^3)Sqrt[1-(x+1/x^2)^2]\[DifferentialD]x

7. ∫(1+ln x)Sqrt[1-x^2(ln x)^2]\[DifferentialD]x

8. ∫x^x Sqrt[x ln x](1+ln x)\[DifferentialD]x

Note: My professor said that #8 will include a function called Erfi[x]. I have no idea what that is.

Again, any help is appreciated.

For #3, I've reduced it, with some algebra, to $\frac{1}{4i}\int\frac{(xe^{ix}-xe^{-ix})(ln(x)+ln(e^{ix}-e^{-ix})-ln(2i))((x-i)e^{ix}+(x+i)e^{-ix})}{1-\sqrt{1-(\frac{1}{2}-\frac{i}{2})\sqrt{xe^{ix}-xe^{-ix}}}}dx$. However, it looks a little...complex. [/lolsofunnymathpunsftw]

I think a u-sub of some complex exponential might be good, but I can't be sure until I try.

 

[zared619]

Thanks for all the help so far. I really appreciate it. This isn't due until Friday in U.S time, but I will try some of your suggestions.