Question

What is `int (-2x^3-x ) / (-2x^2+x +7 )`?

Answer 1

`x^2/2+x/2+17/8ln|-2x^2+x+7|+45/(8sqrt57)ln|(4x-1-sqrt57)/(4x-1+sqrt57)|+C.`

Explanation:

Let, `I=int(-2x^3-x)/(-2x^2+x+7)dx`

We have, `-2x^3-x=x(-2x^2+x+7)+1/2(-2x^2+x+7)-17/2x-7/2.`

`:.(-2x^3-x)/(-2x^2+x+7)=x+1/2+(-17x/2-7/2)/(-2x^2+x+7).`

Also, `d/dx(-2x^2+x+7)=-4x+1," we set, "-17/2x-7/2=17/8(-4x+1)-45/8.`

Thus, we have,

`(-2x^3-x)/(-2x^2+x+7)=x+1/2+{17/8d/dx(-2x^2+x+7)-45/8}/(-2x^2+x+7).`

`:. I=int[x+1/2+{17/8d/dx(-2x^2+x+7)-45/8}/(-2x^2+x+7)]dx,`

`=x^2/2+x/2+17/8ln|-2x^2+x+7|+45/8int1/(2x^2-x-7)dx,`

`=x^2/2+x/2+17/8ln|-2x^2+x+7|+45/16int1/(x^2-x/2-7/2)dx,`

`=x^2/2+x/2+17/8ln|-2x^2+x+7|+45/16intdx/{(x-1/4)^2-57/16),`

`=x^2/2+x/2+17/8ln|-2x^2+x+7|`

`+45/16*1/(2(sqrt57/4))ln|(x-1/4-sqrt57/4)/(x-1/4+sqrt57/4)|,`

`rArr I=x^2/2+x/2+17/8ln|-2x^2+x+7|`

`+45/(8sqrt57)ln|(4x-1-sqrt57)/(4x-1+sqrt57)|+C.`

Enjoy Maths.!