Which of these is equal to \(\int\frac{10x-11}{x^2+x-2}\text{?}\)
\(\displaystyle 7\ln|x-1|+3\arctan(x+2)+C\)
\(\displaystyle 7\ln|x-1|+3\ln|x+2|+C\)
\(\displaystyle 7\arctan(x-1)+3\arctan(x+2)+C\)
\(\displaystyle 7\arctan(x-1)+3\ln|x+2|+C\)
Observation5.6.7.
To find integrals like \(\int \frac{x^2+x+1}{x^3+x} dx\) and \(\int\frac{10x-11}{x^2+x-2}\text{,}\) we'd like to decompose the fractions into simpler partial fractions that may be integrated with these formulas
Quadratic Terms: Let \((x^2+bx+c)^n\) divide \(q(x)\text{,}\) where \(x^2+bx+c\) is irreducable. Then the decomposition of \(\frac{p(x)}{q(x)}\) will contain the terms
By using the form of the decomposition \(\displaystyle \frac{x^2+5x+3}{(x+1)^2x}=\frac{A}{x+1}+\frac{B}{(x+1)^2}+\frac{C}{x}\) and the coefficients found in Activity 5.6.13 and Activity 5.6.14, evaluate \(\displaystyle \int \frac{x^2+5x+3}{(x+1)^2x} dx\text{.}\)
Activity5.6.16.
Given that \(\displaystyle\frac{x^3-7x^2-7x+15}{x^3(x+5)}=\frac{A}{x}+\frac{B}{x^2}+ \frac{C}{x^3}+\frac{D}{x+5}\) do the following to find \(A, B, C\text{,}\) and \(D\text{.}\)
Plug in an \(x\) value that lets you find the value of \(C\text{.}\)
(c)
Plug in an \(x\) value that lets you find the value of \(D\text{.}\)
(d)
Use other algebra techniques to find the values of \(A\) and \(B\text{.}\)
Activity5.6.17.
Given your choice in Activity 5.6.16 Find \(\displaystyle\int \frac{x^3-7x^2-7x+15}{x^3(x+5)} dx.\)
Activity5.6.18.
Consider the rational expression \(\displaystyle\frac{2x^3+2x+4}{x^4+2x^3+4x^2}.\) Which of the following is the partial fraction decomposition of this rational expression?
Given your choice in Activity 5.6.18 Find \(\displaystyle\int \frac{2x^3+2x+4}{x^4+2x^3+4x^2} dx\text{.}\)
Activity5.6.20.
Given that \(\displaystyle \frac{2x+5}{x^2+3x+2}=\frac{-1}{x+2}+\frac{3}{x+1}\text{,}\) find \(\displaystyle\int_0^3 \frac{2x+5}{x^2+3x+2} dx\text{.}\)