Antiderivative of Log X

Introduction to anti-derivative of log x:

The natural logarithm is the logarithm to the base e, where e is an irrational constant approximately equal to 2.718. The natural logarithm is generally written as ln(x), loge(x) or sometimes, if the base of e is implicit, as simply log(x). Formally, ln(a) may be defined as the area under the graph of `1/x ` from 1 to a, that is as the anti-derivatives or integral,

ln a  = `int_1^a(1/x)dx`

This defines a logarithm because it satisfies the fundamental property of a logarithm:

ln (ab) = ln a + ln b

Source Wikipedia.

Anti-derivative Logarithmic Formulas:

1. `int` `(1/x)` dx = log x + c

2.  `int` e x dx = e x + c

3. `int (dx) / (a^2 - x^2) ` = `(1/(2a)) log [(a + x) / (a - x)] + c`

4. `int (dx) / (x^2 - a^2)` = `(1/(2a)) log [(x - a) / (x + a)] + c`

5. `int (dx) / sqrt(a^2 - x^2)` = `sin^-1(x / a) + c`

6. `int (dx) / sqrt(x^2 - a^2) ` = `log [(x + sqrt(x^2 - a^2)] + c`

Anti-derivative Logarithmic Problems:

Anti-derivative logarithmic problem 1:

Find the anti-derivative of given logarithmic function, log x  with respect to x

Solution:

Given logarithmic function, ` int ` log x. dx

Let,    u = log x                            dv = dx.

`(du)/(dx) ` = `1/x`                                    v = x

du = `1/x` dx

We know anti-derivative parts formula,  `int ` u dv = uv - `int ` v du

`int ` log x. dx  = log x . x - `int`` x ((dx) /x) `

= x. log x - `int` dx

= x. log x - x + c

= x( log x - 1) + c

Answer: Anti-derivative of log x is    x( log x - 1) + c

Anti-derivative logarithmic problem 2:

Find the anti-derivative of given logarithmic function, `(1 + 25x)/x^2`   with respect to x

Solution:

Given function, ` int` `(1 + 25x)/x^2` . dx

`int``(1 + 25x)/x^2`. dx  =` int` `dx/x^2` + `int` ` (25x)/x^2 ` dx            

= `int` `x^(-2) ` dx + ` int `` 25x^(-1)` dx               

= `x^(-1)` +  25 log x + c

Answer:  Anti-derivative of  `(1 + 25x)/x^2`  is x-1 + 25 log x + c

Anti-derivative logarithmic problem 3:

Find the anti-derivative of given logarithmic function, `e^(3x)/(1- e^(3x))` with respect to x

Solution:

Let u = 1- e3x           du = - `3` `e^(3x)` dx      

So, substitute the u and du

`int `  `e^(3x)/(1- e^(3x))` dx = `int`` (-1/3)(du)/u`                                 

= `(-1/3)` ` int` `1/u` du

= `(-1)/3` ln u + c                                                      we know u = 1- e3x

=` (- ln (1-e^(2x)))/3` +c

= `((-1)/3)` ln(e3x -1) + c

Answer:  Anti-derivative of `e^(3x)/(1- e^(3x))` is  `((-1)/3)` ln(e3x -1) + c