∫ [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!
â« (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) â« (u-3)du /u
=(1/25) â« [ 1-3/u] du
=u/25 -(3/25) ln|u| +C
=(3+5x) / 25 - (3/25) ln }3+5x| +C
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Verified answer
∫ [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!
â« (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) â« (u-3)du /u
=(1/25) â« [ 1-3/u] du
=u/25 -(3/25) ln|u| +C
=(3+5x) / 25 - (3/25) ln }3+5x| +C