Example of integrals involving quadratics, Mathematics

Assignment Help:

Evaluate the following integral.

∫√(x2+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,

X2+4x+5 = x2+4x+4-4+5=(x+2)2+1

After completing the square the integral becomes like this:

∫√(x2 + 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 = sec2 θdθ

√((x + 2)2 +1)

= √ tan2 θ+1

=√ sec2 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,

20_Example of Integrals Involving Quadratics 2.png

∫ √x2 + 4x + 5 dx = ∫ sec3 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 = √(x2 + 4x +5/1)

        = √ (x2+4x+5)

1954_Example of Integrals Involving Quadratics 1.png

∫ √(x2+4x+5) dx = ½ ((x+2)√x2+4x+5+1n|x+2+√x2+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.


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