Math help. simple 10 points!

what is the intergral of (0.5) / (x + 1) ??
i understand it would be .5 ln (x + 1).

alternatively.... isnt (.5) / (x+ 1) the same as 1 / (2x + 2)???? correct me if i am wrong, but isnt the intergral of this now 0.5 ln (2x + 2)
what am i doing wrong?? thanks alot

Both integrals are correct, strangely. Try it- take the derivatives of both, and you will get the same thing.

Isn't this impossible?

Nope. Turns out that When you take the derivative of a logarithm, the coefficient inside does not matter. Why, you ask? (let D[ function] = its derivative)

D[ ln(ax) ] = D [ ln(x) ]

Why is this true? well, if you remember the properties of the logarithm...

ln(a *b) = ln(a) + ln(b).

In this case, ln(a *x) = ln(a) + ln(x)

But ln(a) is a constant! so when you take the derivative, it is zero.

Therefore, both of the equations that you found are correct.

Hope this helps!

You are correct, except you have not accounted for C.

INT [.5/(x+1)]dx = .5*INT[ 1/(x+1)]= .5 ln|x+1| +C

And INT [1/(2x+2)]dx= (1/2) INT[ 2/(2x+2)] dx= (1/2) ln| 2x+2| +C

But ln|2x+2|= ln2 +ln|x+1|

So allowing (1/2) ln2 to join the C, they are the same.

In other words, you would not write an indefinite integral as x+5+C, only as x+C

Hoping this helps!