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If f((3x-4)/(3x+4))=x+2, then int f(x)dx...

If `f((3x-4)/(3x+4))=x+2`, then `int` f(x)dx is equal to

A

`e^(x+2)log|(3x-4)/(3x+4)|+C`

B

`-(8)/(3)log|1-x|+(2)/(3)x+C`

C

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

D

`e^([(3x-4)//(3x+4)])-(x^(2))/(2)-2x+C`

Text Solution

Verified by Experts

The correct Answer is:
B
Given, `f((3x-4)/(3x+4))=x+2`
Let`" "(3x-4)/(3x+4)=X`
`rArr" "(3x-4+3x+4)/(3x-4-(3x+4))=(X+1)/(X-1)`
`" [By componendo and dividendo]"`
`rArr" "(6x)/(-8)=(X+1)/(X-1)" "rArr" "x=(4(X+1))/(3(1-X))`
`therefore" "f(x)=(4(X+1))/(3(1-X))+2=(10-2X)/(3(1-X))`
Now, `intf(x)dx=int(10-2x)/(3(1-x))dx=(2)/(3)int(5-x)/(1-x)dx`
`=(2)/(3)[intdx+int(4)/(1-x)dx]`
`=(2)/(3)[x-4log(1-x)]+C`
`=(2)/(3)x-(8)/(3)log(1-x)+C`
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