# Evaluate Integrals Involving Quadratic Expressions

Using Completing Square

Tutorials with examples and detailed solutions and exercises with answers on how to use the techniques of completing square and substitution to evaluate integrals involving quadratic expressions. ## Formulas
We first review some of the derivatives formulas for known inverse functions involving quadratic expressions.
We now use the above differentiation formulas to write integrals as follows. NOTE: in what follows, K is the constant of integration. Finally one more integral is computed using partial fractions decomposition as follows 5)
\int \dfrac{1}{{1 - x^2}} dx = \int (\dfrac{1/2}{x+1}- \dfrac{1/2}{x-1}) dx = \dfrac{1}{2} \ln \dfrac{x+1}{x-1} + K
## Examples## Example 1Evaluate the integral
\int \dfrac{1}{\sqrt{-x^2 - x}} dx
= \int \dfrac{1}{\sqrt{1/4 - (x+1/2)^2}} dx =
Factor out 1/4 from under the square root
= \int \dfrac{2}{\sqrt{1 - (2(x+1/2))^2}} dx
Let z = 2(x + 1/2) = 2x + 1 and therefore dz/2 = dx and the integral becomes
= \int \dfrac{2}{\sqrt{1 - z^2}} dz
= arcsin(z) + K = arcsin(2x + 1) + K
## Example 2Evaluate the integral
\int \dfrac{2}{3x^2 + 12x + 24} dx
= (1 / 6) \int \dfrac{1}{(x/2 + 1)^2 + 1} dx
Let z = x/2 + 1 and therefore dx = 2dz and rewrite the integral as
= (1 / 3) \int \dfrac{1}{z^2 + 1} dz
= (1/3) arctan(z) + K
= (1/3) arctan(x/2 + 1) + K
## Example 3Evaluate the integral
\int \dfrac{1}{\sqrt{x^2 + 12x + 40}} dx
= \int \dfrac{1}{\sqrt{( x + 6 )^2 + 4}} dx
Factor 4 out from under the square root
= \int \dfrac{1}{2\sqrt{( x/2+ 3 )^2 + 1}} dx
Let z = x/2 + 3, hence 2 dz = dx, and the integral may be written
= \int \dfrac{1}{\sqrt{z^2 + 1}} dz
= arcsinh(z) + K = arcsinh(x/2 + 3) + K
## ExercisesEvaluate the integrals given below1.
\int \dfrac{3}{\sqrt{9 - x^2}} dx
2.
\int \dfrac{3}{x^2 + 12x + 45} dx
3.
\int \dfrac{\sqrt2}{\sqrt{2x^2 + 10x + 13}} dx
## Answers to Above Exercises1. 3 arcsin(x / 3) + K 2. arctan(x/3 + 2) + K 3. arcsinh(2x + 5) ## More References and linksintegrals and their applications in calculus. |