Integral with completing the square

Question

∫ (dx)/(sqrt(x^2+4x+13)) 

 

Im having a really hard time solving this, if you can show me steps I would appreciate it. Thanks 

Answer 1

Try rewriting x² + 4x + 13 by completing the square.  It should simplify to something that you can integrate.

Answer 2

If you complete the square, you get x² + 4x + 13 = x² + 4x + 4 + 9 = (x+2)² + 3².  But you probably already knew that.  At this point, a substitution will simplify it.  If we select 3u = x + 2 or u = (x+2)/3, then 3du = dx and our integral becomes:

 

∫ 3du / √(9*(u² + 1)) = ∫ du / √(u² + 1)

 

At this point, you'll need to pull from a table of integrals (or know from memory) that the antiderivative is:

 

sinh^-1(u) + C

 

All that's left to do is re-substitute the initial value of u:

 

sinh^-1((x+2)/3) + C

 

If you have additional questions, feel free to let me know.