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int(1)/(x^(2)-9)dx is equal to...

`int(1)/(x^(2)-9)dx` is equal to

A

`log|(x-3)/(x+3)|+C`

B

`(1)/(6)log|(x-3)/(x+3)|+C`

C

`(1)/(6)log|(x+3)/(x-3)|+C`

D

`log|(x+3)/(x-3)|+C`

Text Solution

Verified by Experts

The correct Answer is:
B
`int(1)/(x^(2)-9)dx=int(1)/(x^(2)-3^(2))dx=int(1)/((x+3)(x-3))dx`
Let`" "(1)/((x+3)(x-3))=(A)/((x+3))+(B)/((x-3))`
`" "1=A(x-3)+B(x+3)`
`rArr" "1=x(A+B)+(-3A+3B)`
On equation the coefficients of x and constant term on both sides, we get
`A+B=0 and -3A+3B=1`
On solving, we get `A=-(1)/(6) and B=(1)/(6)`
`therefore int(1)/((x+3)(x-3))dx=int((-1))/(6(x+3))dx+int(1)/(6(x-3))dx`
`=-(1)/(6)log|x+3|+(1)/(6)log|x-3|+C`
`=(1)/(5)log|(x-3)/(x+3)|+C`
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