# Integrate

• Apr 24th 2013, 12:07 PM
jdesak
Integrate
How to integrate this? I = (10x-2x^2)/[(x-1)^2 * (x+3)] dx
• Apr 24th 2013, 12:42 PM
ebaines
Re: Integrate
Hint: convert the expression using partial fraction expansion into three separate fractions:

$\frac {10x-2x^2}{(x-1)^2(x+3)} = \frac A {(x-1)^2} + \frac B {x-1} + \frac C {x+3}$

Determine values for A, B and C and then it's easy to integrate.
• Apr 24th 2013, 03:53 PM
x3bnm
Re: Integrate
By partial fraction expansion we know that:

\begin{align*} \frac A {(x-1)^2} + \frac B {x-1} + \frac C {x+3} = \frac {10x-2x^2}{(x-1)^2(x+3)} \\A(x+3) + B(x+3)(x-1) + C(x-1)^2 =& -2x^2 +10x\\ (B+C)x^{2} +(A +2B - 2C)x + (3A - 3B + C) =& -2x^2 +10x\end{align*}

\begin{align*}B + C =& -2.....\text{(1)}\\ A + 2B - 2C =& 10........\text{(2)}\\ 3A - 3B + C =& 0..........\text{(3)}\end{align*}

Solving these three equations we get:

$\therefore\,\, A = 2, \,\, B = 1 \,\, \text{and } C = -3$

Now:

\begin{align*}\int \frac {10x-2x^2}{(x-1)^2(x+3)}\,\,dx =& \int \left(\frac{2}{(x-1)^2} + \frac{1}{x-1} - \frac{3}{x+3}\right)\,\,dx\\=& \int \frac{2}{(x-1)^2}\,\,dx + \int \frac{1}{x-1}\,\,dx - \int \frac{3}{x+3}\,\,dx....\text{(4)}\end{align*}

$\text{Let } u = x - 1 \therefore\,\, du = dx$

$\text{ and let } v = x + 3 \therefore\,\, dv = dx$

\begin{align*}\int \frac {10x-2x^2}{(x-1)^{2}(x+3)}\,\,dx =& \int \frac{2}{u^2}\,\,du +\int \frac{1}{u}\,\,du -\int\frac{3}{v}\,\,dv.............\text{[by plugging in eq.4 }u, v, du, dv\text{]}\\ =& \frac{-2}{u} + \ln{(u)} - 3\ln{(v)} + C\\ =& \frac{-2}{x-1} + \ln{(x-1)} - 3\ln{(x+3)} + C...\text{[by plugging in values of } u \text{ and } v \text{]}\end{align*}
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 8, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}