Explain integration and logs (1 Viewer)

[Makro]

I don't understand the formulae given. Could someone explain it simply to me? I seem to have done the work once but don't understand it. An example or two would be nice. Thank you!

 

[kurt.physics]

I don't understand the formulae given. Could someone explain it simply to me? I seem to have done the work once but don't understand it. An example or two would be nice. Thank you!

which formulae exactly?

 

[Makro]

I don't understand either of them or when to use either of them.

 

[study-freak]

I don't understand either of them or when to use either of them.

When to use: when you see (derivative of denominator)/(denominator) inside an integral.
What it is: y=log (to base e) x (I'll call it ln x) is an inverse function of y=e^x.
Consider y=e^x
change x and y
x=e^y
y=ln x. That's the definition of it.
In graphical terms, it is a reflection of y=e^x on the line y=x.

Now, let y=lnx...(1)
e^y=x
d(e^y)/dy=dx/dy
e^y=dx/dy
dy/dx=e^(-y)(reciprocal)
=e^(-lnx) using (1)
=e^(lnx^(-1)) using log laws
=x^(-1)
=1/x.

hence


is just an extension of above to all function f(x).

 

[Makro]

So I just wrote all that out on paper and it appears to make sense.

y = lnx
y' = 1/x

And I now understand the function over the function thing, I was a bit confused before. So we've got:


using the formula, I'd say the answer is ln (x²+2)

but there's a 1/2. I want to know where all these fractions are coming from as they appear on most questions.

 

[Fortify]

The rule is f'(x)/f(x) = ln f(x).

So we have x/(x^2 + 2), we need to make the top f'(x) which is 2x.

So 1/2 int 2x / (x^2 + 2) is the same as int x / (x^2 + 2) right?

So 1/2 int 2x / (x^2 + 2) satisfies the rule f'(x)/f(x) = ln f(x).

So 1/2 ln f(x) + C or 1/2 ln (x^2 + 2) + C.

 

[study-freak]

derivative of (x^2+2)=2x
x/(x^2+2)=1/2*2x/(x^2+2)
and now apply the formula, giving 1/2*ln(x^2+2)+c

 

[Makro]

I think it's all making sense. Thanks a bunch to those who posted I'll post back if I fail later.