What is the integral of (x-8)/(x^2-7x+10) ?

The integral of (x-8)/(x^2-7x+10) is 2(1/4 ln|4x^2-28x+40|+3/4 (ln|(2x-7)/3+1|-ln|(2x-7)/3-1|))+C

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integral of (x-8)/(x^2-7x+10)

\int \frac{x - 8}{x ^{2} - 7 x + 10} d x

2 (\frac{1}{4} ln 4 x^{2} - 28 x + 40 + \frac{3}{4} (ln \frac{2 x - 7}{3} + 1 - ln \frac{2 x - 7}{3} - 1)) + C

Solution steps

\int \frac{x - 8}{x ^{2} - 7 x + 10} d x

Apply u-substitution

: \int \frac{u}{u ^{2} + 9 u + 18} d u

= \int \frac{u}{u ^{2} + 9 u + 18} d u

Complete the square

u^{2} + 9 u + 18 : (u + \frac{9}{2})^{2} - \frac{9}{4}

= \int \frac{u}{( u + \frac{9}{2} ) ^{2} - \frac{9}{4}} d u

Apply u-substitution

: \int \frac{2 ( 2 v - 9 )}{4 v ^{2} - 9} d v

= \int \frac{2 ( 2 v - 9 )}{4 v ^{2} - 9} d v

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int \frac{2 v - 9}{4 v ^{2} - 9} d v

Expand

\frac{2 v - 9}{4 v ^{2} - 9} : \frac{2 v}{4 v ^{2} - 9} - \frac{9}{4 v ^{2} - 9}

Apply the Sum Rule:

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

= 2 (\int \frac{2 v}{4 v ^{2} - 9} d v - \int \frac{9}{4 v ^{2} - 9} d v)

\int \frac{2 v}{4 v ^{2} - 9} d v = \frac{1}{4} ln 4 v^{2} - 9

\int \frac{9}{4 v ^{2} - 9} d v = - \frac{3}{4} (ln \frac{2}{3} v + 1 - ln \frac{2}{3} v - 1)

= 2 (\frac{1}{4} ln 4 v^{2} - 9 - (- \frac{3}{4} (ln \frac{2}{3} v + 1 - ln \frac{2}{3} v - 1)))

Substitute back

= 2 (\frac{1}{4} ln 4 (x - 8 + \frac{9}{2})^{2} - 9 - (- \frac{3}{4} (ln \frac{2}{3} (x - 8 + \frac{9}{2}) + 1 - ln \frac{2}{3} (x - 8 + \frac{9}{2}) - 1)))

Simplify

2 (\frac{1}{4} ln 4 (x - 8 + \frac{9}{2})^{2} - 9 - (- \frac{3}{4} (ln \frac{2}{3} (x - 8 + \frac{9}{2}) + 1 - ln \frac{2}{3} (x - 8 + \frac{9}{2}) - 1))) : 2 (\frac{1}{4} ln 4 x^{2} - 28 x + 40 + \frac{3}{4} (ln \frac{2 x - 7}{3} + 1 - ln \frac{2 x - 7}{3} - 1))

= 2 (\frac{1}{4} ln 4 x^{2} - 28 x + 40 + \frac{3}{4} (ln \frac{2 x - 7}{3} + 1 - ln \frac{2 x - 7}{3} - 1))

Add a constant to the solution

= 2 (\frac{1}{4} ln 4 x^{2} - 28 x + 40 + \frac{3}{4} (ln \frac{2 x - 7}{3} + 1 - ln \frac{2 x - 7}{3} - 1)) + C

x \to 2 lim (x 20 - x^{2})

2 x y^{^{'}} = 2 x e^{x} - 2 y + 6 x^{2}

x \to 74 lim (\frac{( 1 )}{76 x + 71})

arcsec (x^{8})

\int e^{x} \cdot 3 x^{2} d x

What is the integral of (x-8)/(x^2-7x+10) ?
The integral of (x-8)/(x^2-7x+10) is 2(1/4 ln|4x^2-28x+40|+3/4 (ln|(2x-7)/3+1|-ln|(2x-7)/3-1|))+C