Can't understand

[Timon]

Hi All,
Sorry for a simple question, but could somebody help? In the example below, after Integral 12x^3dx there 3x^4+(1/2)*c. I couldn't understand why 1/2? it should be just C.
Thank you.

 

[Dr.Peterson]

I would guess that they called it 1/2 c knowing they would immediately be multiplying by 2.

If they had just called it c, then the next step would have 2c, and they would want to replace that with a new constant, say k = 2c, just to avoid unnecessary details in the final answer.

It's a smart thing to do oneself, but a poor thing to do when you want to teach someone else and don't bother explaining why you did it.

 

[Steven G]

Since c can be any number, then so can 1/2*c and 7+c or ln(c). There are many ways to write a constant. In differential equations you will learn that ln(x) + c will usually be written as ln(cx). Do you see why?