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. |