Evaluate the following integral.

∫√(x^{2}+4x+5) dx

**Solution:**

Remind from the Trig Substitution section that to do a trig substitution here we first required to complete the square on the quadratic. This provides,

X^{2}+4x+5 = x2+4x+4-4+5=(x+2)^{2}+1

After completing the square the integral becomes like this:

∫√(x^{2} + 4x +5) dx

= ∫ √ ((x+2)^{2} 1dx)

Upon doing this we can recognize the trig substitution that we require. Here it is,

x + 2 = tan θ

x= tan θ -2

dx = sec^{2} θdθ

√((x + 2)^{2} +1)

= √ tan^{2} θ+1

=√ sec^{2} θ

=|sec θ |

= sec θ

Recall that as we are doing an indefinite integral we be able to drop the absolute value bars. By using this substitution the integral becomes,

∫ √x^{2} + 4x + 5 dx = ∫ sec^{3} θ d θ

= ½ (secθ tanθ + ln |secθ + tan θ|) + c

We can end the integral out along with the following right triangle.

tanθ = (x+2/1)

secθ = √(x^{2} + 4x +5/1)

= √ (x^{2}+4x+5)

∫ √(x^{2}+4x+5) dx = ½ ((x+2)√x^{2}+4x+5+1n|x+2+√x^{2}+4x+5|) + c

Thus, by completing the square we were capable to take an integral that had a general quadratic in it and transform it into a form that permitted us to make use of a known integration technique.