Question

integration of (x^3+8)(x-1)/x^2-2x+4​

Answer 1

$\large\underline{\textsf{\textbf{\purple{ \mapsto Given:}}}}$

• $\sf \displaystyle\int\dfrac{(x^3-8)(x-1)}{x^2-2x+4}$

$\large\underline{\textsf{\textbf{\purple{ \mapsto To\:Find:}}}}$

• To find its integration.

$\large\underline{\textsf{\textbf{\purple{ \mapsto Answer:}}}}$

$\sf \pink{We \;have}$

$\tt\red{\longmapsto}\displaystyle\tt\int\dfrac{(x^3-8)(x-1)}{x^2-2x+4}$

$\tt\red{\longmapsto}\displaystyle\tt\int \dfrac{(x+2)\cancel{(x^2-2x+4)(x-1)}}{\cancel{x^2-2x+4}}$

$\tt\red{\longmapsto}\displaystyle\tt\int \dfrac{(x-2)(x-1)}{1}$

$\tt\red{\longmapsto}\displaystyle\tt\int (x+2)(x-1)$

$\tt\red{\longmapsto}\displaystyle\tt\int x(x-1)+2(x-1)$

$\tt\red{\longmapsto}\displaystyle\tt\int x^2-x+2x-2$

$\tt\red{\longmapsto}\displaystyle\tt\int x^2+x-2$

$\tt\red{\longmapsto}\displaystyle\tt\int x^2+\int x -\int 2x^0$

$\boxed{\boxed{\green{\bf \orange{\dag}\:\int x^n=\dfrac{x^{n+1}}{n+1}}}}$

$\tt\red{\longmapsto} \dfrac{x^{2-1}}{2+1}+\dfrac{x^{1+1}}{1+1}+\dfrac{x^{0+1}}{0+1}$

$\underline{\boxed{\blue{\tt\red{\leadsto} \dfrac{x^3}{3}+\dfrac{x^2}{2}-2x+c}}}$