Find the indefinite integral by susbtitution `int dx/(1+5x)`

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Find `int (dx)/(1+5x)` :

Let u=1+5x. Then du=5dx. Multiply the integrand by 5 and 1/5:

`int(dx)/(1+5x)=int (1/5*5*dx)/(1+5x)`

`=1/5int(5dx)/(1+5x)`

`=1/5int (du)/u`

`=1/5ln|u|+C` Substitute u=1+5x:

`=1/5ln|1+5x|+C`

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`int(dx)/(1+5x)=1/5ln|1+5x|+C`

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Check: `d/(dx)[1/5ln(1+5x)+C]`

`=d/(dx)1/5ln(1+5x)`

`=1/5d/(dx)ln(1+5x)`

`=1/5(5/(1+5x))`

`=1/(1+5x)`

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