Find ∫f(x)dx

If,

f[(3x-4)/3x+4)] = x+2

Dear Shreyans

first calculate f(x)

put 3x-4/3x+4 = y

calculate x in terms of y

x=4(1+y)/3(1-y)

x+2=2(5-y)/3(1-y) = 2(y-5)/3(y-1)

f(y)=2(y-5)/3(y-1)

f(x)=2(x-5)/3(x-1)

to integrate this use partial fractions

f(x)=2/3 [(x-1-4)/(x-1)]= 2/3[1 - 4/x-1]

now integration is simple

∫2/3[1 - 4/x-1]dx

2/3[x-4log(x-1)] + C

All the best.                                                           

AKASH GOYAL

AskiitiansExpert-IIT Delhi

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put 3x-4/3x+4=t,

on solving by componendo and dividendo,

6x/-8=t+1/t-1

x=4(t+1)/3(1-t)

put this in the given condition, get f(x), then integrate it