Evaluate: `int(3x-2)/((x+1)^2(x+3)) dx`

1.	Evaluate: `int(3x-2)/((x+1)^2(x+3)) dx`
Answer» `Let ((3x-2))/((x+1)^(2)(x+3))=(A)/((x+1))+(B)/((x+1)^(2))+(C)/((x+3))` `implies (3x-2)-=A(x+1)(x+3)+B(x+3)+C(x+1)^(2).` Putting `x=-3` on both sides of (i) , we get `C=(-11)/(4).` Putting `x=-1` on both sides of (i), we get `B=(-5)/(2).` Comparing the coefficients of `x^(2)` on both sides of (i) ,we get `A+C=0impliesA=-C=(11)/(4).` `therefore ((3x-2))/((x+1)^(2)(x+3))=(11)/(4(x+1))-(5)/(2(x+1)^(2))-(11)/(4(x+3))` `implies int((3x-2))/((x+1)^(2)(x+3))dx=(11)/(4).int (dx)/((x+1))-(5)/(2).int (1)/((x+1)^(2))dx-(11)/(4).int(dx)/((x+3))` `=(11)/(4).log \|x+1\|+(5)/(2(x+1))-(11)/(4).log\|x+3\|+C` `=(11)/(4)/log\|(x+1)/(x+3)\|+(5)/(2(x+1))+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`