Solution to Example 1:
We use partial fractions decomposition to decompose the integrand into simpler fractions.
\(
\dfrac{-5x + 11}{x^2+x-2} = \dfrac{2}{x-1} - \dfrac{7}{x+2}
\)
Solution to Example 2:
A partial fractions decomposition of the integrand gives
\(
\dfrac{x^2+6x - 3}{(x+3)(x^2+2x+9)} = \dfrac{2x+2}{x^2+2x+9} - \dfrac{1}{x+3}
\)
We now use a table of integrals to evaluate the integrals
\(
\displaystyle \int \dfrac{x^2+6x - 3}{(x+3)(x^2+2x+9)} dx = \int \dfrac{2x+2}{x^2+2x+9} dx - \int \dfrac{1}{x+3} dx \\
= \ln|x^2+2x+9| - \ln|x+3| + C
\)
Solution to Example 3:
In this example the degree of the numerator is greater that the degree of the denominator and therefore a division of the numerator by the denominator is carried out in order to write the integrand as follows:
\(
\dfrac{2x^3 + 10x^2 +11x}{x^2+5x+6} = 2x - \dfrac{x}{x^2+5x+6}
\)
A partial fractions decomposition of the term \( \dfrac{x}{x^2+5x+6} \) gives
\(
\dfrac{2x^3 + 10x^2 +11x}{x^2+5x+6} = 2x - \dfrac{x}{x^2+5x+6} = 2x +\dfrac{2}{x+2}-\dfrac{3}{x+3}
\)
Using the above, the given integral may be written as
\(
\displaystyle \int \dfrac{2x^3 + 10x^2 +11x}{x^2+5x+6} dx = \int 2x dx +\int \dfrac{2}{x+2} dx - \int \dfrac{3}{x+3} dx
\)
>Using a table of integrals, we evaluate the integrals as follows:
\(
= x^2 + 2 \ln|x+2| - 3 \ln|x+3| + C
\)
Questions
Evaluate the following integrals.
1.
\(
\displaystyle \int \dfrac{-x + 7}{x^2+x-2} dx
\)
