How do I evaluate ∫ (x)/(3+5x) dx?

Question

How do I evaluate ∫ (x)/(3+5x) dx?

Answer 1

∫ [x /(3 + 5x)] dx =

it is a rational integrand whose numerator and denominator are of the same

degree, whereas the degree of the numerator should be lower;

therefore, in order to reduce its degree, do as follows:

divide and multiply the integral by 5:

(1/5) ∫ [5x /(3 + 5x)] dx =

add (3 - 3) to the numerator:

(1/5) ∫ [(5x + 3 - 3)/(3 + 5x)] dx =

break it up into:

(1/5) ∫ {[(5x + 3)/(3 + 5x)] - [3/(3 + 5x)]} dx =

(1/5) ∫ [(5x + 3)/(3 + 5x)] dx - (1/5) ∫ [3/(3 + 5x)] dx =

simplify the first integral:

(1/5) ∫ dx - (1/5) ∫ [3/(3 + 5x)] dx =

(1/5)x - (1/5) ∫ [3/(3 + 5x)] dx =

as for the remaining integral, attempting to change the numerator into the

derivative of the denominator, divide and multiply by (5/3):

(1/5)x - (1/5)(3/5) ∫ {[(5/3)3]/(3 + 5x)} dx =

(1/5)x - (3/25) ∫ [5/(3 + 5x)] dx =

that is:

(1/5)x - (3/25) ∫ [d(3 + 5x)]/ (3 + 5x) =

(1/5)x - (3/25) ln | 3 + 5x | + C

in conclusion:

∫ [x /(3 + 5x)] dx = (1/5)x - (3/25) ln | 3 + 5x | + C

I hope it helps..

Bye!

Answer 2

∫ (x)/(3+5x) dx

let 3+5x=u

5x=u-3

x=(u-3)/5

5 dx = du

dx =(1/5) du

The given integral becomes

(1/5)(1/5) ∫ (u-3)du /u

=(1/25)  ÃƒÂ¢Ã‚ˆÂ« [ 1-3/u] du

=u/25 -(3/25) ln|u| +C

=(3+5x) / 25 - (3/25) ln }3+5x| +C