## 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, ∫ √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) ∫ √(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.

#### Problem solving, compare 643,251;633,512; and 633,893. The answer is 633,51...

compare 643,251;633,512; and 633,893. The answer is 633,512

#### identity function, detail on identity functio

detail on identity function

#### Calculate the volume of rectangular piece of cardboard, 1. A rectangular pi...

1. A rectangular piece of cardboard measuring 15 inches by 24 inches is to be made into a box with an open top by cutting equal size squares from each comer and folding up the side

#### Heat lost in a cylindrical pipe, solution for this project

solution for this project

#### Calculate the profit the bank earn each treasury bond, Financial institutio...

Financial institutions often create synthetic instruments out of existing instruments.  In this case an investment bank plans to buy Treasury Bonds with 20-year maturities at their

#### Define a hamilton path, Define a Hamilton path. Determine if the following ...

Define a Hamilton path. Determine if the following graph has a Hamilton circuit. Ans: A path is known as a Hamiltonian path if it consists of every vertex of the graph e

#### Simplify the logical expression, Simplify the logical expression X‾ Y‾ + X‾...

Simplify the logical expression X‾ Y‾ + X‾ Z + Y Z +Y‾ Z W‾  Ans: The K-Map for the following Boolean expression is described by the following diagram. The optimized expression

#### Which of the subsequent numbers will yield a number larger, Which of the su...

Which of the subsequent numbers will yield a number larger than 23.4 while it is multiplied by 23.4? When multiplying through a number less than 1, you get a product in which i

#### Algorithm for division, Also, their inability to apply the algorithm for di...

Also, their inability to apply the algorithm for division becomes quite evident. The reason for these difficulties may be many. We have listed some of them below. 1) There are n

#### I am an expert of maths, i want to be get hired, please guide me.

i want to be get hired, please guide me.  