Integrate: dx/sqrt(x^2+2x +5)

Homework Statement

Integrate: dx/sqrt(x^2+2x +5)

refer to above

The Attempt at a Solution

I can integrate the equation dx/sqrt(1+x^2) using the rules

cosh^2 u - sinh^2 u = 1
cosh^2 u = 1+sinh^2 u

but i dont know where to start with this question because of the 2x + 5 in the denomentor. Could someone please point me in the right direction.

George Jones
Staff Emeritus
Gold Member
Could someone please point me in the right direction.

Complete the square.

Complete the square.

Thanks that helped but i'm still stuck with a 4 i don't know to get rid of :S

integral of: dx/sqrt(x^2+2x+5)

Equals the integral of: dx/sqrt((x+1)^2 + 4)

Using:
1. (x+1) = sinh u
2. cosh^2 u = 1 + sinh^2 u
3. dx = cosh u du

I get to this stage:

Integral of: dx/sqrt((x+1)^2 + 4)

Equals the integral of: cosh u du/((x+1)^2 + 4)

This is where i get stuck, i'm not sure what to do with the 4. could som1 plz help.

thanx again George

George Jones
Staff Emeritus
Gold Member
In an appropriate manner, take the 4 outside the square root.

In an appropriate manner, take the 4 outside the square root.

ya i squared everything but the bottom line is not in the correct form.

i hav:

cosh ^2 u du / (x+1)^2 +4

if the denomentator was (x+1)^2 + 1 i could just substitute but cos of that 4 i can't. gettin rid of taht 4 is my problem.

George Jones
Staff Emeritus