Question

integration of x⁵-4⁴-3x³-2x²+2x+5/ x⁴+x³

Answer 1

Given to evaluate,

$\displaystyle\sf{\longrightarrow I=\int\dfrac {x^5-4x^4-3x^3-2x^2+2x+5}{x^4+x^3}\ dx}$

$\displaystyle\sf{\longrightarrow I=\int\dfrac {x^5+x^4-5x^4-5x^3+2x^3-2x^2+2x+5}{x^4+x^3}\ dx}$

$\displaystyle\sf{\longrightarrow I=\int\dfrac {x(x^4+x^3)-5(x^4+x^3)+2x^3-2x^2+2x+5}{x^4+x^3}\ dx}$

$\displaystyle\sf{\longrightarrow I=\int\dfrac {(x^4+x^3)(x-5)+(2x^3-2x^2+2x+5)}{x^4+x^3}\ dx}$

$\displaystyle\sf{\longrightarrow I=\int\left(x-5+\dfrac {2x^3-2x^2+2x+5}{x^4+x^3}\right)\ dx}$

$\displaystyle\sf{\longrightarrow I=\int (x-5)\ dx+\int\dfrac {2x^3-2x^2+2x+5}{x^4+x^3}\ dx}$

$\displaystyle\sf{\longrightarrow I=\dfrac {1}{2}\,x^2-5x+\int\dfrac {2x^3-2x^2+2x+5}{x(x+1)(x^2-x+1)}\ dx\quad\quad\dots(1)}$

Let,

$\displaystyle\sf{\longrightarrow \dfrac {2x^3-2x^2+2x+5}{x(x+1)(x^2-x+1)}=\dfrac {A}{x}+\dfrac {B}{x+1}+\dfrac {2Cx-D}{x^2-x+1}}$

$\displaystyle\sf{\longrightarrow 2x^3-2x^2+2x+5=A(x+1)(x^2-x+1)+Bx(x^2-x+1)+x(2Cx-D)(x+1)}$

$\displaystyle\sf{\longrightarrow 2x^3-2x^2+2x+5=(A+B+2C)x^3-(B+D-2C)x^2+(B-D)x+A}$

Equating the corresponding coefficients,

• $\displaystyle\sf {A+B+2C=2}$
• $\displaystyle\sf {B+D-2C=2}$
• $\displaystyle\sf {B-D=2}$
• $\displaystyle\sf {A=5}$

Solving them we get,

• $\displaystyle\sf {A=5}$
• $\displaystyle\sf {B=\dfrac {1}{3}}$
• $\displaystyle\sf {C=D=-\dfrac {5}{3}}$

Thus (1) becomes,

$\displaystyle\sf{\longrightarrow I=\dfrac {1}{2}\,x^2-5x+5\int\dfrac {1}{x}\ dx+\dfrac {1}{3}\int\dfrac {1}{x+1}\ dx-\dfrac {5}{3}\int\dfrac {2x-1}{x^2-x+1}\ dx}$

$\displaystyle\sf{\longrightarrow\underline {\underline {I=\dfrac {1}{2}\,x^2-5x+5\log|x|+\dfrac {1}{3}\log|x+1|-\dfrac {5}{3}\log(x^2-x+1)+C}}}$