Question

1/5x-3,Integrate the given function defined on proper domain w.r.t. x.

Answer 1

In the attachment I have answered this problem.   I have applied substitution method to find derivative of the given function    See the attachment for detailed solution.

Answer 2

we have to find the value of $\int{\frac{1}{5x-3}}\,dx$

we know, $\int{\frac{f'(x)}{f(x)}}\,dx=lnf(x)+C$

here, f(x) = (5x - 3)
and f'(x) = d(5x- 3)/dx = 5

so, $\int{\frac{1}{5x-3}}\,dx=\int{\frac{1}{5}\frac{5}{5x-3}}\,dx$

now, 5/(5x - 3) seems like f'(x)/f(x)

hence, $\int{\frac{1}{5}\frac{5}{5x-3}}\,dx$

= $\frac{1}{5}ln(5x-3)+C$