Integrating $\int\frac{1}{kx}dx$

Question

$\int\frac{1}{kx}dx$

There are two ways to integrate this...

Method 1 (Separating the coefficient from the variable):

$\frac{1}{k}\int\frac{1}{x}dx$

$\frac{\ln|x|}{k} + c$

Method 2 (knowing that it is an ln() function and multiplying by the reciprocal of the coefficient):

$\int\frac{1}{kx}dx$

$\frac{\ln|kx|}{k}+c$

When both solutions are derived, we get the same answer of $\frac{1}{kx}$.

Which solution is the correct solution?

Since, as you write "When both solutions are derived, we get the same answer of $\frac{1}{kx}$", they are both right. — MasB, May 02 '16 at 23:52

score 4 · Accepted Answer · edited May 03 '16 at 00:12

4

$${\frac{\ln(kx)}{k}+C = \frac{\ln(k)}{k}+\frac{\ln(x)}{k}+C = \frac{\ln(x)}{k} + \tilde{C}}$$

So they're the same, they just differ by a constant

edited May 03 '16 at 00:12

TakamaruTaihou

answered May 02 '16 at 23:50

Brenton

Oh! It never occurred to me that more than one answer could be right... This is eye-opening. – TakamaruTaihou May 02 '16 at 23:58
@TakamaruTaihou technically indefinite integrals have infinite answers. – Oion Akif Oct 11 '16 at 17:39

score 2 · Answer 2 · answered May 02 '16 at 23:50

2

Both are fine, they differ by a constant.

and actually $$\int \frac{1}{x} dx = \ln |x| $$

answered May 02 '16 at 23:50

Siong Thye Goh

2 Answers2