Evaluate using long division.
The integral of x/6x-5 dx
How do I divide this?
forgotten your Algebra I?
x/(6x-5) = 1/6 + 5/6 * 1/(6x-5)
the way I read that is that you first want to divide x by 6x-5, and then integrate that series
using long division ....
x/(6x-5) = 1/6 + 5/(36x) + 25/(216x^2) + 125/(1296x^3 + ..
(a geometric series with a = 1/6 and r = 5/(6x)
so ∫x/(6x-5) dx
= ∫(1/6 + 5/(36x) + 25/(216x^2) + ...) dx
= (1/6)x + (5/36)lnx - 25/(216x) - 125/(2592x^2) - .....
To divide the given polynomial using long division, follow these steps:
1. Make sure both the dividend (numerator) and divisor (denominator) are written in descending order of exponents.
In this case, the dividend is x, and the divisor is 6x - 5.
2. Divide the highest-degree term of the dividend by the highest-degree term of the divisor.
The highest-degree term of the dividend is x, and the highest-degree term of the divisor is 6x. So the result is x/6x = 1/6.
3. Multiply the divisor by the result from step 2.
Multiply (6x - 5) by 1/6, which gives (6x - 5)/6.
4. Subtract the result from step 3 from the dividend.
Subtract (6x - 5)/6 from x.
x + 5/6
____________________
6x - 5 | x
- (6x - 5)/6
__________________
5. Bring down the next term from the dividend.
Since there are no other terms in the dividend, the process is complete.
The quotient is x/6x = 1/6, and the remainder is 5/6.