What is the integral of (4x-5)/(4x^2-12x+13) ?

The integral of (4x-5)/(4x^2-12x+13) is 1/2 ln|4x^2-12x+13|+1/4 arctan(x-3/2)+C

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integral of (4x-5)/(4x^2-12x+13)

\int \frac{4 x - 5}{4 x ^{2} - 12 x + 13} d x

\frac{1}{2} ln 4 x^{2} - 12 x + 13 + \frac{1}{4} arctan (x - \frac{3}{2}) + C

Solution steps

\int \frac{4 x - 5}{4 x ^{2} - 12 x + 13} d x

Expand

\frac{4 x - 5}{4 x ^{2} - 12 x + 13} : \frac{4 x}{4 x ^{2} - 12 x + 13} - \frac{5}{4 x ^{2} - 12 x + 13}

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{4 x}{4 x ^{2} - 12 x + 13} d x - \int \frac{5}{4 x ^{2} - 12 x + 13} d x

\int \frac{4 x}{4 x ^{2} - 12 x + 13} d x = 2 (\frac{1}{4} ln 4 x^{2} - 12 x + 13 + \frac{3}{4} arctan (x - \frac{3}{2}))

\int \frac{5}{4 x ^{2} - 12 x + 13} d x = \frac{5}{4} arctan (x - \frac{3}{2})

= 2 (\frac{1}{4} ln 4 x^{2} - 12 x + 13 + \frac{3}{4} arctan (x - \frac{3}{2})) - \frac{5}{4} arctan (x - \frac{3}{2})

Simplify

2 (\frac{1}{4} ln 4 x^{2} - 12 x + 13 + \frac{3}{4} arctan (x - \frac{3}{2})) - \frac{5}{4} arctan (x - \frac{3}{2}) : \frac{1}{2} ln 4 x^{2} - 12 x + 13 + \frac{1}{4} arctan (x - \frac{3}{2})

= \frac{1}{2} ln 4 x^{2} - 12 x + 13 + \frac{1}{4} arctan (x - \frac{3}{2})

Add a constant to the solution

= \frac{1}{2} ln 4 x^{2} - 12 x + 13 + \frac{1}{4} arctan (x - \frac{3}{2}) + C

\int \frac{e ^{- l n (1 - x^{2})}}{x ^{2}} d x

\frac{d y}{d x} = \frac{( x - 2 )}{y + 1}

\frac{2 x ^{2} + 6 x + 6}{x}

\int x 8 - x^{2} d x

(- 1, 19), (5, - 41)

What is the integral of (4x-5)/(4x^2-12x+13) ?
The integral of (4x-5)/(4x^2-12x+13) is 1/2 ln|4x^2-12x+13|+1/4 arctan(x-3/2)+C