Mastering Tricky Antiderivatives for Exams: Integrals with Helpful Hints

[csi86]

I have to prepare for the exams and this is a set of integrals I can't do... some hints pls :

1.$$\int \frac{x}{\sqrt{x^{2}+2x+2}} \; dx$$

2.$$\int \frac{x}{x^{3}+1} \; dx$$


Thank you for your time.

 

[siddharth]

Hi and welcome to PF csi86!

You need to show some work before you receive help. What are your thoughts/ideas on these problem? What have you done till now, and where are you stuck?

 

[dextercioby]

For the second try to solve

$$\frac{x}{x^{3}+1} =\frac{A}{x+1} +\frac{Bx+C}{x^{2}-x+1}$$

Daniel.

 

[courtrigrad]

For the first one use integration by parts:

$$u = x$$$$dv = \frac{1}{\sqrt{1+(x+1)^{2}}}\; dx$$

 

[csi86]

Thank you, I have succesfully solved the second one, I might have done some mistakes :
$$-ln|x+1| + \frac{1}{2} ln{(x^{2}-x+1)} + \frac{2}{\sqrt{3}}\arctan{\frac{2x-1}{\sqrt{3}}}$$

 

[dextercioby]

Just don't forget the integration constant.

Daniel.

 

[csi86]

Indeed , this might hurt in an exam.

I tried to solve the first one and I end up with :
$$\frac{x^{2}}{2 \sqrt{x^{2}+2x+2}} + \frac{1}{2} \int \frac{(x^{2})(2x+2)}{2 \sqrt{(x^{2}+2x+2})^{3}}} \;dx$$

(Using integration by parts)

 

[dextercioby]

Write it like that

$$\int \frac{x}{\sqrt{x^{2}+2x+2}} \ dx =\frac{1}{2}\int \frac{d(x^{2}+2x+2)}{\sqrt{x^{2}+2x+2}} -\int \frac{dx}{\sqrt{x^{2}+2x+2}}$$

Daniel.

 

[csi86]

I figured it out, thanks for the help Daniel. :)

I end up with :
$$\sqrt{x^{2}+2x+2} - \ln{(x+1+ \sqrt{x^{2}+2x+2})}$$

I don't know if that is correct but I honestly hope so :).