What is the integral of (1-2x)/(x^2+9x+14) ?

The integral of (1-2x)/(x^2+9x+14) is -ln|4x^2+36x+56|+2ln| 1/5 (-2x-9)+1|-2ln| 1/5 (-2x-9)-1|+C

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integral of (1-2x)/(x^2+9x+14)

\int \frac{1 - 2 x}{x ^{2} + 9 x + 14} d x

- ln 4 x^{2} + 36 x + 56 + 2 ln \frac{1}{5} (- 2 x - 9) + 1 - 2 ln \frac{1}{5} (- 2 x - 9) - 1 + C

Solution steps

\int \frac{1 - 2 x}{x ^{2} + 9 x + 14} d x

Apply u-substitution

: \int - \frac{2 u}{u ^{2} - 20 u + 75} d u

= \int - \frac{2 u}{u ^{2} - 20 u + 75} d u

Take the constant out:

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

= - 2 \cdot \int \frac{u}{u ^{2} - 20 u + 75} d u

Complete the square

u^{2} - 20 u + 75 : (u - 10)^{2} - 25

= - 2 \cdot \int \frac{u}{( u - 10 ) ^{2} - 25} d u

Apply u-substitution

: \int \frac{v + 10}{v ^{2} - 25} d v

= - 2 \cdot \int \frac{v + 10}{v ^{2} - 25} d v

Expand

\frac{v + 10}{v ^{2} - 25} : \frac{v}{v ^{2} - 25} + \frac{10}{v ^{2} - 25}

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{v}{v ^{2} - 25} d v + \int \frac{10}{v ^{2} - 25} d v)

\int \frac{v}{v ^{2} - 25} d v = \frac{1}{2} ln v^{2} - 25

\int \frac{10}{v ^{2} - 25} d v = - ln \frac{v}{5} + 1 + ln \frac{v}{5} - 1

= - 2 (\frac{1}{2} ln v^{2} - 25 - ln \frac{v}{5} + 1 + ln \frac{v}{5} - 1)

Substitute back

= - 2 (\frac{1}{2} ln (1 - 2 x - 10)^{2} - 25 - ln \frac{1 - 2 x - 10}{5} + 1 + ln \frac{1 - 2 x - 10}{5} - 1)

Simplify

- 2 (\frac{1}{2} ln (1 - 2 x - 10)^{2} - 25 - ln \frac{1 - 2 x - 10}{5} + 1 + ln \frac{1 - 2 x - 10}{5} - 1) : - ln 4 x^{2} + 36 x + 56 + 2 ln \frac{1}{5} (- 2 x - 9) + 1 - 2 ln \frac{1}{5} (- 2 x - 9) - 1

= - ln 4 x^{2} + 36 x + 56 + 2 ln \frac{1}{5} (- 2 x - 9) + 1 - 2 ln \frac{1}{5} (- 2 x - 9) - 1

Add a constant to the solution

= - ln 4 x^{2} + 36 x + 56 + 2 ln \frac{1}{5} (- 2 x - 9) + 1 - 2 ln \frac{1}{5} (- 2 x - 9) - 1 + C

\int \frac{- e ^{x}}{1 - e ^{2 x}} d x

\int 2 e^{- x - 2 y} d y

\int \frac{6 x ^{2} + 4 x + 2}{x ^{3} + x ^{2} + x} d x

\int (cos^{2} (\frac{x}{2})) d x

What is the integral of (1-2x)/(x^2+9x+14) ?
The integral of (1-2x)/(x^2+9x+14) is -ln|4x^2+36x+56|+2ln| 1/5 (-2x-9)+1|-2ln| 1/5 (-2x-9)-1|+C