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#Help wihere the random constant come out?
narrow nova
clear prawnBOT
pine finch
is this the answer that was presented to you or is part of the development of the answer?
narrow nova
answer that is presented to me
they integrated 1/ax+b and got that
i was wondering where the 1/a lnk came from
even if it did cancel out
this is the question
pine finch
the only thing that i can think of is that they used the literal antiderivative of 1/(ax+b)
narrow nova
well i mean it could just be 1/(ax+b+c) or something
narrow nova
the only thing that i can think of is that they used the literal antiderivative ...
wdym
pine finch
that is (1/a)(ln(ax+b) +C)
pine finch
they just set the constant to look like ln(k)
then distributed the 1/a inside the parentesis
pine finch
well its 1/a ln|ax+b| + c
well yeah
the book author chose to write the constant in a different way
ln(k) = C
narrow nova
and what about 1/a?
pine finch
that is (1/a)(ln(ax+b) +C)
form this
when you solve for the antiderivative you get a 1/a factor
so you get (1/a)C = (1/a)ln(k)
narrow nova
when you solve for the antiderivative you get a 1/a factor
i mean when you evaluate it its (1/a)ln(ax+b) +C but ig you can take out 1/a
narrow nova
well if you differentiate (1/a)ln(ax+b) +C you get 1/ax+b
narrow nova
it's actually (1/a)(ln(ax+b) +C)
wait how?
pine finch
you set the integral of 1/(ax+b)
narrow nova
ok
pine finch
then you multiply by a a/a factor so you get 1/a times integral of a/(ax+b)
narrow nova
?
pine finch
so you have in the integral argument the derivative of the ln(ax+b)
then you solve the integral wich get you the ln(|ax+b|) and then you multiply this by the 1/a factor
now this is the kind of not skiping any steps way
pine finch
i guess my book isnt that rigorous
in your book they just merged the (1/a)ln(k) into one constant since it doesn't really matter
narrow nova
i mean its the same thing just your way is more rigorous but my way in the textbook is more intuitive?
pine finch
so in many books or developments they just leav it as C
narrow nova
ok ok
narrow nova
then you multiply by a a/a factor so you get 1/a times integral of a/(ax+b)
yep, btw i dont get this
pine finch
the derivative of ln(ax+b)=a/(ax+b)
narrow nova
ok
pine finch
in the ontegral we have just 1/(ax+b)
narrow nova
ah ok i see
narrow nova
fair, cya
pine finch
bye
thick flint
the example you provided is a definite integral