What is the integral of (12-9z)/(1-4z-4z^2) ?

The integral of (12-9z)/(1-4z-4z^2) is

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integral of (12-9z)/(1-4z-4z^2)

\int \frac{12 - 9 z}{1 - 4 z - 4 z ^{2}} d z

- \frac{9}{2} (- \frac{1}{4} ln - 324 z^{2} - 324 z + 81 + \frac{1}{12 2} (11 ln - \frac{2 z + 1}{2} + 1 - 11 ln \frac{2 z + 1}{2} + 1)) + C

Solution steps

\int \frac{12 - 9 z}{1 - 4 z - 4 z ^{2}} d z

Apply u-substitution

: \int - \frac{9 u}{- 4 u ^{2} + 132 u - 927} d u

= \int - \frac{9 u}{- 4 u ^{2} + 132 u - 927} d u

Take the constant out:

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

= - 9 \cdot \int \frac{u}{- 4 u ^{2} + 132 u - 927} d u

Complete the square

- 4 u^{2} + 132 u - 927 : - 4 (u - \frac{33}{2})^{2} + 162

= - 9 \cdot \int \frac{u}{- 4 ( u - \frac{33}{2} ) ^{2} + 162} d u

Apply u-substitution

: \int \frac{2 v + 33}{2 ( - 4 v ^{2} + 162 )} d v

= - 9 \cdot \int \frac{2 v + 33}{2 ( - 4 v ^{2} + 162 )} d v

Take the constant out:

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

= - 9 \cdot \frac{1}{2} \cdot \int \frac{2 v + 33}{- 4 v ^{2} + 162} d v

Expand

\frac{2 v + 33}{- 4 v ^{2} + 162} : \frac{2 v}{- 4 v ^{2} + 162} + \frac{33}{- 4 v ^{2} + 162}

Apply the Sum Rule:

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

= - 9 \cdot \frac{1}{2} (\int \frac{2 v}{- 4 v ^{2} + 162} d v + \int \frac{33}{- 4 v ^{2} + 162} d v)

\int \frac{2 v}{- 4 v ^{2} + 162} d v = - \frac{1}{4} ln - 4 v^{2} + 162

\int \frac{33}{- 4 v ^{2} + 162} d v = \frac{11}{12 2} (ln \frac{2 v}{9} + 1 - ln \frac{2 v}{9} - 1)

= - 9 \cdot \frac{1}{2} (- \frac{1}{4} ln - 4 v^{2} + 162 + \frac{11}{12 2} (ln \frac{2 v}{9} + 1 - ln \frac{2 v}{9} - 1))

Substitute back

= - 9 \cdot \frac{1}{2} (- \frac{1}{4} ln - 4 (12 - 9 z - \frac{33}{2})^{2} + 162 + \frac{11}{12 2} (ln \frac{2 ( 12 - 9 z - \frac{33}{2} )}{9} + 1 - ln \frac{2 ( 12 - 9 z - \frac{33}{2} )}{9} - 1))

Simplify

- 9 \cdot \frac{1}{2} (- \frac{1}{4} ln - 4 (12 - 9 z - \frac{33}{2})^{2} + 162 + \frac{11}{12 2} (ln \frac{2 ( 12 - 9 z - \frac{33}{2} )}{9} + 1 - ln \frac{2 ( 12 - 9 z - \frac{33}{2} )}{9} - 1)) : - \frac{9}{2} (- \frac{1}{4} ln - 324 z^{2} - 324 z + 81 + \frac{1}{12 2} (11 ln - \frac{2 z + 1}{2} + 1 - 11 ln \frac{2 z + 1}{2} + 1))

= - \frac{9}{2} (- \frac{1}{4} ln - 324 z^{2} - 324 z + 81 + \frac{1}{12 2} (11 ln - \frac{2 z + 1}{2} + 1 - 11 ln \frac{2 z + 1}{2} + 1))

Add a constant to the solution

= - \frac{9}{2} (- \frac{1}{4} ln - 324 z^{2} - 324 z + 81 + \frac{1}{12 2} (11 ln - \frac{2 z + 1}{2} + 1 - 11 ln \frac{2 z + 1}{2} + 1)) + C

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What is the integral of (12-9z)/(1-4z-4z^2) ?
The integral of (12-9z)/(1-4z-4z^2) is