What is f(x) = int x/(x-1) dx if f(2) = 0 ?

1 Answer
May 26, 2016

Since ln can't help you, set the denominator because of its simple form as a variable. When you solve the integral, just set x=2 to fit the f(2) in the equation and find the integration constant.

Answer is:

f(x)=x+ln|x-1|-2

Explanation:

f(x)=intx/(x-1)dx

The ln function will not help in this case. However, since the denominator is quite simple (1st grade):

Set u=x-1=>x=u+1

and (du)/dx=d(x+1)/dx=(x+1)'=1=>(du)/dx=1<=>du=dx

intx/(x-1)dx=int(u+1)/(u)du=int(u/u+1/u)du=

=int(1+1/u)du=int1du+int(du)/u=u+ln|u|+c

Substituting x back:

u+ln|u|+c=x-1+ln|x-1|+c

So:

f(x)=intx/(x-1)dx=x-1+ln|x-1|+c

f(x)=x-1+ln|x-1|+c

To find c we set x=2

f(2)=2-1+ln|2-1|+c

0=1+ln1+c

c=-1

Finally:

f(x)=x-1+ln|x-1|+c=x-1+ln|x-1|-1=x+ln|x-1|-2

f(x)=x+ln|x-1|-2