# Integration by substitution of a lineal function

I notice that the integral of a linear function gives different results by using a method or another

Integrating directly,

$$\int (a+bx) dx = ax+b\dfrac{x^2}{2} +c_1$$

And making a substitution, $$\begin{split} \int (a+bx) dx &\\ u=(a+bx) &\Rightarrow du=bdx \\ \int u \dfrac{du}{b} = \dfrac{1}{b}\dfrac{u^2}{2} +c_2 &=\dfrac{a^2}{2b}+\dfrac{bx^2}{2}+ax \end{split}$$

And comparing results, they are obviously not equal

$$ax+b\dfrac{x^2}{2} +c_1 = ax +\dfrac{bx^2}{2} + \dfrac{a^2}{2b} +c_2$$

My question is, how to explain that? Is it simply an arbitrary integration constants problem?

• Yes, it just has to do with the constants. $$c_1=c_2+\frac{a^2}{2b}$$ – Franklin Pezzuti Dyer Sep 26 '17 at 23:19
• They're identical up to changing the constant. But an indefinite integral is only determined up to an arbitrary constant anyway, so that it appears to differ when using different techniques is basically irrelevant. – Chappers Sep 26 '17 at 23:29