# Basic integration but two different answers?

#### powerhouseteam

Greetings all

1/(x/2 + 1) dx

I get the answer 2 ln (x/2 + 1)

my friend did it this way

1/(x/2 + 1) * (2/2) dx
2/(x+2) dx

2 ln (x+2)

What is going on??

#### mr fantastic

MHF Hall of Fame
Greetings all

1/(x/2 + 1) dx

I get the answer 2 ln (x/2 + 1)

my friend did it this way

1/(x/2 + 1) * (2/2) dx
2/(x+2) dx

2 ln (x+2)

What is going on??
What's going on is that both your answers are wrong - for two reasons. One of the reasons explains why you can get two apprently different answers.

C and K are arbitrary constants of integration. Your teacher will have told you many many times not to forget including it.

Note: 2 ln |x/2 + 1| + C = 2 ln |1/2 (x + 2)| + C = 2 ln (1/2) + 2 ln|x+2| + C = 2 ln|x + 2| + K where K = C + 2ln(1/2).

#### chiph588@

MHF Hall of Honor
Greetings all

1/(x/2 + 1) dx

I get the answer 2 ln (x/2 + 1)

my friend did it this way

1/(x/2 + 1) * (2/2) dx
2/(x+2) dx

2 ln (x+2)

What is going on??
$$\displaystyle 2\ln|\tfrac x2+1|$$+C$$\displaystyle = 2\ln|\tfrac x2+1|+2\ln(2)+C_1 = 2\ln|(\tfrac x2+1)\cdot2|+C_1$$ where the last equality come from the law $$\displaystyle \ln(x)+\ln(y)=\ln(xy)$$
Thus we see $$\displaystyle 2\ln|\tfrac x2+1|+C = 2\ln|x+2|+C_1$$