Question

How do you use Integration by Substitution to find `intdx/(5-3x)`?

Answer 1

The answer is

`-1/3*ln(5-3x)+c`, where c is constant

Solution
For problems like

`int dx/(a+bx)`,

we start with assuming `a+bx=u`
then, differentiating this assumption `b*dx=du`

`dx=(du)/b`

Now, substituting this in problem,

`int(du)/(b*u) = 1/b*lnu +c`, where c is constant

now, substituting u in the solution,

`1/b*ln(a+bx)+c`

Similarly following for the problem,
let `5-3x=u`
then, `-3*dx=du`
now substituting in the problem, we get

`int-1/3*(du)/u = -1/3*ln(u)+c`

Finally, plugging in u, the answer will be `-1/3*ln(5-3x)+c`