What is the integral of 1/(x^2+12x+35) ?

The integral of 1/(x^2+12x+35) is 1/2 ln|x+5|-1/2 ln|x+7|+C

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integral of 1/(x^2+12x+35)

\int \frac{1}{x ^{2} + 12 x + 35} d x

\frac{1}{2} ln ∣ x + 5 ∣ - \frac{1}{2} ln ∣ x + 7 ∣ + C

Solution steps

\int \frac{1}{x ^{2} + 12 x + 35} d x

Take the partial fraction of

\frac{1}{x ^{2} + 12 x + 35} : \frac{1}{2 ( x + 5 )} - \frac{1}{2 ( x + 7 )}

= \int \frac{1}{2 ( x + 5 )} - \frac{1}{2 ( x + 7 )} d x

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{1}{2 ( x + 5 )} d x - \int \frac{1}{2 ( x + 7 )} d x

\int \frac{1}{2 ( x + 5 )} d x = \frac{1}{2} ln ∣ x + 5 ∣

\int \frac{1}{2 ( x + 7 )} d x = \frac{1}{2} ln ∣ x + 7 ∣

= \frac{1}{2} ln ∣ x + 5 ∣ - \frac{1}{2} ln ∣ x + 7 ∣

Add a constant to the solution

= \frac{1}{2} ln ∣ x + 5 ∣ - \frac{1}{2} ln ∣ x + 7 ∣ + C

(- 4.4) (- 1.5)

\frac{d}{d x} (\frac{9}{sin ( x ) + cos ( x )})

(- 3, \frac{2}{5}), (4, \frac{3}{10})

h \to 0 lim (\frac{\frac{1}{1 + h} - 1}{h})

What is the integral of 1/(x^2+12x+35) ?
The integral of 1/(x^2+12x+35) is 1/2 ln|x+5|-1/2 ln|x+7|+C