# 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.
Review:
\dfrac{d(\arcsin(x))}{dx} = \dfrac{1}{\sqrt{1 - x^2}}
2)
\dfrac{d(\arctan(x))}{dx} = \dfrac{1}{{1 + x^2}}
3)
\dfrac{d(\text{arcsinh}(x))}{dx} = \dfrac{1}{\sqrt{1 + x^2}}
4)
\dfrac{d(\text{arccosh}(x))}{dx} = \dfrac{1}{\sqrt{x^2 - 1}}
We now use the above differentiation formulas to write integrals as follows. NOTE: in what follows, K is the constant of integration. 1)
\int \dfrac{1}{\sqrt{1 - x^2}} dx = \arcsin(x) + K
2)
\int \dfrac{1}{{1 + x^2}} dx = \arctan(x) + K
3)
\int \dfrac{1}{\sqrt{1 + x^2}} dx = \text{arcsinh(x)} + K
4)
\int \dfrac{1}{\sqrt{x^2 - 1}} dx = \text{arccosh(x)} + K
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
\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
\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
\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
\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 on integrals and their applications in calculus. |