What is the integral of ((6x-12))/(x^2-8x+25) ?

The integral of ((6x-12))/(x^2-8x+25) is 4arctan(1/3 (x-4))+3ln|x^2-8x+25|+C

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integral of ((6x-12))/(x^2-8x+25)

\int \frac{( 6 x - 12 )}{x ^{2} - 8 x + 25} d x

4 arctan (\frac{1}{3} (x - 4)) + 3 ln x^{2} - 8 x + 25 + C

Solution steps

\int \frac{6 x - 12}{x ^{2} - 8 x + 25} d x

Apply Integration By Parts

: 2 arctan (\frac{1}{3} (x - 4)) (x - 2) - \int 2 arctan (\frac{1}{3} (x - 4)) d x

= 2 arctan (\frac{1}{3} (x - 4)) (x - 2) - \int 2 arctan (\frac{1}{3} (x - 4)) d x

\int 2 arctan (\frac{1}{3} (x - 4)) d x = 2 x arctan (\frac{1}{3} (x - 4)) - 3 ln x^{2} - 8 x + 25 - 8 arctan (\frac{1}{3} (x - 4))

= 2 arctan (\frac{1}{3} (x - 4)) (x - 2) - (2 x arctan (\frac{1}{3} (x - 4)) - 3 ln x^{2} - 8 x + 25 - 8 arctan (\frac{1}{3} (x - 4)))

Simplify

2 arctan (\frac{1}{3} (x - 4)) (x - 2) - (2 x arctan (\frac{1}{3} (x - 4)) - 3 ln x^{2} - 8 x + 25 - 8 arctan (\frac{1}{3} (x - 4))) : 4 arctan (\frac{1}{3} (x - 4)) + 3 ln x^{2} - 8 x + 25

= 4 arctan (\frac{1}{3} (x - 4)) + 3 ln x^{2} - 8 x + 25

Add a constant to the solution

= 4 arctan (\frac{1}{3} (x - 4)) + 3 ln x^{2} - 8 x + 25 + C

\int \frac{x ^{3}}{4 + 5 x ^{4}} d x

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\int \frac{12}{x ^{5}} + \frac{1}{4 x ^{8}} + \frac{6}{7 x ^{2}} d x

What is the integral of ((6x-12))/(x^2-8x+25) ?
The integral of ((6x-12))/(x^2-8x+25) is 4arctan(1/3 (x-4))+3ln|x^2-8x+25|+C