Question

integrate (1-2x)/(45+4x-x^2) dx without using trig substitution

Answer 1

$x^2-4x-45=0$

$x=\frac{4\pm \sqrt{16+180}}{2}$

$x_1=-5,x_2=9$

$x^2-4x-45=(x+5)(x-9)$

$\frac{1-2x}{45+4x-x^2}=\frac{2x-1}{(x+5)(x-9)}=\frac{A}{x+5}+\frac{B}{x-9}$

$A(x-9)+B(x+5)=2x-1$

$A+B=2$

$5B-9A=-1$

$5(2-A)-9A=-1$

$A=11/14,B=17/14$

$\int\frac{1-2x}{45+4x-x^2}dx=\int(\frac{11}{14(x+5)}+\frac{17}{14(x-9)})dx=\\=\frac{11}{14}log(x+5)+\frac{17}{14}log(x-9)+C$