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Integration of logarithmic function : If a function 'f' is of the form f(x) = $x^{-1}$ gives us the result in the absolute value of the natural log function, as shown below.1) $\int_{}^{} x^{-1}dx = \int_{}^{}\frac{1}{x}dx = ln|x| + c$

2) $\int_{}^{} ln(x) dx = x ln|x| - x + c $

3) $\int_{}^{} \log_{a}{x} dx = \frac{x}{ln a}ln(x-1) + c $

Use the rule for integration of logarithmic function,

$\int_{}^{}\frac{4}{x} dx$ = 4 ln|x| + c

$\int_{}^{}\frac{4}{x} dx = ln(x^{4}) + c $ ( use the property of logarithm)

**Example 2: ** Integrate $\frac{5}{x-5}$ with respect to x.

**Solution : ** $\int_{}^{}\frac{5}{x-5} dx = 5 \int_{}^{}\frac{1}{x-5} dx $

Use the 'U' substitution method,

Let x - 5 = u

dx = du

$5 \int_{}^{}\frac{1}{x-5} dx = 5 \int_{}^{}\frac{1}{u} du$

Use the integration of logarithmic function rule

= 5 ln|u|

Now plug in u = x - 5

$ \int_{}^{}\frac{5}{x-5} dx = 5 ln|x-5| + c $

**Example 3: ** Integrate $\frac{x}{x^{2} +1}$ with respect to x.

**Solution : ** $\int_{}^{}\frac{x}{x^{2} +1} dx $

Use the 'U' substitution method,

Let $x^{2}$ + 1 = u

2x dx = du

x dx = $\frac{1}{2}$ du

$ \int_{}^{}\frac{x}{x^{2} +1} dx = \int_{}^{}\frac{1}{u} \frac{1}{2} du$

= $\frac{1}{2} \int_{}^{}\frac{1}{u} du $

Use the integration of logarithmic function rule

= $\frac{1}{2} ln|u|$

Now plug in u = $x^{2} + 1 $

$\int_{}^{}\frac{x}{x^{2} +1} dx = \frac{1}{2} ln|x^{2} +1| + c $

**Example 4: ** Integrate $\frac{sin(x)}{cos(x) -3 }$ with respect to x.

**Solution : ** $\int_{}^{}\frac{sin(x)}{cos(x) -3 } dx $

Use the 'U' substitution method,

Let cos(x) - 3 = u

-sin(x) dx = du

sin(x) dx = - du

$ \int_{}^{}\frac{sin(x)}{cos(x) -3 } dx = \int_{}^{}\frac{-1}{u} du$

= $-1\int_{}^{}\frac{1}{u} du $

Use the integration of logarithmic function rule

= $- ln|u|$

Now plug in u = $cos(x) -3 $

$\int_{}^{}\frac{sin(x)}{cos(x) -3 }dx = - ln|cos(x)-3 | + c $

**Example 5: ** Integrate $4\log_{3}{x}$ with respect to x.

**Solution : ** $\int_{}^{}\log_{3}{x} dx = 4 \int_{}^{}\log_{3}{x} dx $

Use the rule for integration of logarithmic function,

$\int_{}^{} 4\log_{3}{x} dx = \frac{4x}{ln 3}ln(x-1) + c $

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