General Calculus.

Hi all!

I need help on two calculus problems:

Firstly ...

$\displaystyle y = tan^-^1(lnx)$

I know the solution which is $\displaystyle \frac{dy}{dx} = \frac{1}{x} . \frac{1}{1 + (lnx)^2}$

What are the rules to get this problem solved?

Secondly ...

When doing Intergration on partial fraction lets take an example:

$\displaystyle
\int\frac{4x^2 - 24x + 11}{(x-3)^2(x+2)} = \frac{3}{(x+2)} + \frac{1}{(x-3)} - \frac{5}{(x-3)^2}$

Then the integral becomes

$\displaystyle
\int3ln(x+2) + ln(x-3) ... ?$

How can I integrate the last bit: $\displaystyle -\frac{5}{(x-3)^2} $

I hope this will help others too!

Quote:

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Originally Posted by dadon

Hi all!

I need help on two calculus problems:

Firstly ...

$\displaystyle y = tan^-^1(lnx)$

I know the solution which is $\displaystyle \frac{dy}{dx} = \frac{1}{x} . \frac{1}{1 + (lnx)^2}$

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This is chain rule.
You got,
$\displaystyle y=\tan^{-1}(u)$ where $\displaystyle u=\ln x$
Thus,
$\displaystyle \frac{dy}{du}=\frac{1}{1+u^2}$ and
$\displaystyle \frac{du}{dx}=\frac{1}{x}$
Thus,
$\displaystyle \frac{dy}{dx}=\frac{dy}{du}\cdot \frac{du}{dx}$
Thus,
$\displaystyle \frac{dy}{dx}=\frac{1}{x}\cdot \frac{1}{1+u^2}$
But, $\displaystyle u=\ln x$ thus,
$\displaystyle \frac{1}{x}\cdot \frac{1}{1+\ln^2 x}$

Quote:

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Originally Posted by dadon

How can I integrate the last bit: $\displaystyle -\frac{5}{(x-3)^2} $

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You need,
$\displaystyle \int \frac{5}{(x-3)^2}dx$
Let, $\displaystyle u=x-3$
Then, $\displaystyle du=dx$
Thus, use this substitution to get,
$\displaystyle \int 5u^{-2}du=-5u^{-1}+C$
But, $\displaystyle u=x-3$ thus,
$\displaystyle -5(x-3)^{-1}+C$

Thank you! I appreciate your help and nicely explained!

Regards,

dadon