Evaluate: `int(x^4 dx)/((x-1)(x^2+1))`

1.	Evaluate: `int(x^4 dx)/((x-1)(x^2+1))`
Answer» `(x^(4))/((x-1)(x^(2)+1))=(x^(4))/((x^(3)-x^(2)+x-1))=(x+1)+(1)/((x^(3)-x^(2)+x-1))` `implies(x^(4))/((x-1)(x^(2)+1))=(x+1)+(1)/((x-1)(x^(2)+1)).` `Let (1)/((x-1)(x^(2)+1))=(A)/((x-1))+(bx+C)/((x^(2)+1)).then`, `1-=A(x^(2)+1)+(Bx+C)(x-1).` Putting `x=1` in (ii) , we get `A=(1)/(2).` Comparing the coefficients of `x^(2)` on both sides of (ii) , we get `A+B=0implies B=-A=-(1)/(2).` Comparing the constant Terms on both sides on (ii) , we get `A-C=1implies C=(A-1)=((1)/(2)-1)=(-1)/(2).` `therefore (1)/((x-1)(x^(2)+1))=(1)/(2(x-1))+(-(1)/(2)x-(1)/(2))/((x^(2)+1))` `therefore (x^(4))/((x-1)(x^(2)+1))=(x+1)+(1)/(2(x-1))-(1)/(2).((x+1))/((x^(2)+1))` `=int (x^(4))/((x-1)(x^(2)+1))dx=int (x+1)dx=int(x-1)dx+(1)/(2)int(dx)/((x-1))-(1)/(4).int (2x)/((x^(2)+1))dx-(1)/(2)int(dx)/((x^(2)+1))` `=(x^(2))/(2)+x+(1)/(2)log\|x-1\|-(1)/(4)log(x^(2)+1)-(1)/(2)tan^(-1)x+C.`

Discussion

`int ((1+tanx))/((x-logcosx))dx.`
Examples: ` int sec^2x / sqrt ( 16 + tan^2x) dx`A. `log|tanx+sqrt(tan^(2)x+16)|+C`B. `log|x+sqrt(tan^(2)x+16)|+C`C. `log|tan x-sqrt(tan^(2)x+16)|+C`D. None of these
`int (x)/((x^(4)-16))dx=?`A. `(1)/(4)log |(x^(2)+4)/(x^(2)-4)+C`B. `(1)/(16)log |(x^(2)+4)/(x^(2)-4)+C`C. `(1)/(16)log |(x^(2)-4)/(x^(2)+4)+C`D. None of these
`int(x^(2))/(sqrt(x^(6)-1))dx=?`A. `(1)/(2) log |x^(3)+sqrt(x^(6)-1)|+C`B. `(1)/(3) log |x^(3)+sqrt(x^(6)-1)|+C`C. `(1)/(3) log |x^(3)-sqrt(x^(6)-1)|+C`D. None of these
Evaluate:`int(sinx)/(sqrt(4cos^2x-1))dx`A. `-(1)/(2)log |2cosx+sqrt(4cos^(2)x-1)|+C`B. `-(1)/(3)log |2cosx+sqrt(4cos^(2)x-1)|+C`C. `-(1)/(6)log |cosx+sqrt(2cos^(2)x-1)|+C`D. None of these
Evaluate:`int(3_(4x)-x^2)/((x+2)(x-1))dx`
Evaluate `int dx/(x(x^n+1))`
Evaluate: `int(2x+5)/(x^2-x-2) dx`
Evaluate: `int(x^3)/((x-1)(x-2)) dx`
Evaluate :`int(x^2)/((x^2+4)(x^2+9))dx`