# How do I find the integral #intx/(x-6)dx# ?

By Substitution,

Let us look at some details.

by splitting the integrand,

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To integrate ( \int \frac{x}{x-6} , dx ), we perform polynomial long division to simplify the integrand.

Step 1: Perform polynomial long division.

[ \begin{array}{c|c} x & 1 \ \hline x-6 & x \end{array} ]

Step 2: Multiply the divisor ( x - 6 ) by ( 1 ) to obtain ( x - 6 ).

Step 3: Subtract ( x - 6 ) from ( x ) to get ( 6 ).

Step 4: Since the degree of the remainder (which is ( 6 )) is greater than or equal to the degree of the divisor ( x - 6 ), we rewrite ( \frac{x}{x-6} ) as ( 1 + \frac{6}{x-6} ).

Now, we can integrate ( 1 + \frac{6}{x-6} ) separately.

[ \int \frac{x}{x-6} , dx = \int \left(1 + \frac{6}{x-6}\right) , dx ]

Integrating ( 1 ) with respect to ( x ) gives ( x ), and integrating ( \frac{6}{x-6} ) with respect to ( x ) gives ( 6\ln|x-6| + C ).

So,

[ \int \frac{x}{x-6} , dx = x + 6\ln|x-6| + C ]

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