How do you integrate $\frac{2 x^{2} + 4 x + 12}{x^{2} + 7 x + 10}$ $(2x^2+4x+12)/(x^2+7x+10)$ using partial fractions?

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How do you integrate $\frac{2 x^{2} + 4 x + 12}{x^{2} + 7 x + 10}$ $(2x^2+4x+12)/(x^2+7x+10)$ using partial fractions?

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Answer 1 · 2017-03-13T17:42:16

The answer is $= 2 x - 14 ln (| x + 5 |) + 4 ln (| x + 2 |) + C$ $=2x-14ln(|x+5|)+4ln(|x+2|)+C$

Explanation:

The numerator is

$2 x^{2} + 4 x + 12 = 2 (x^{2} + 2 x + 6)$ $2x^2+4x+12=2(x^2+2x+6)$

We perform a long division

$a a a a$ $color(white)(aaaa)$ $x^{2} + 2 x + 6$ $x^2+2x+6$ $a a a a a$ $color(white)(aaaaa)$ $∣$ $|$ $x^{2} + 7 x + 10$ $x^2+7x+10$

$a a a a$ $color(white)(aaaa)$ $x^{2} + 7 x + 10$ $x^2+7x+10$ $a a a a$ $color(white)(aaaa)$ $∣$ $|$ $1$ $1$

$a a a a a a$ $color(white)(aaaaaa)$ $0 - 5 x - 4$ $0-5x-4$

Therefore

$\frac{2 x^{2} + 4 x + 12}{x^{2} + 7 x + 10} = 2 (1 - \frac{5 x + 4}{x^{2} + 7 x + 10})$ $(2x^2+4x+12)/(x^2+7x+10)=2(1-(5x+4)/(x^2+7x+10))$

We factorise the denominator

$x^{2} + 7 x + 10 = (x + 5) (x + 2)$ $x^2+7x+10=(x+5)(x+2)$

We can perform the decomposition into partial fractions

$\frac{5 x + 4}{x^{2} + 7 x + 10} = \frac{5 x + 4}{(x + 5) (x + 2)}$ $(5x+4)/(x^2+7x+10)=(5x+4)/((x+5)(x+2))$

$= \frac{A}{x + 5} + \frac{B}{x + 2} = \frac{A (x + 2) + B (x + 5)}{(x + 5) (x + 2)}$ $=A/(x+5)+B/(x+2)=(A(x+2)+B(x+5))/((x+5)(x+2))$

The denominators are the same, we compare the numerators

$5 x + 4 = A (x + 2) + B (x + 5)$ $5x+4=A(x+2)+B(x+5)$

Let $x = - 5$ $x=-5$ , $\Rightarrow$ $=>$ , $- 21 = - 3 A$ $-21=-3A$ , $\Rightarrow$ $=>$ , $A = 7$ $A=7$

Let $x = - 2$ $x=-2$ , $\Rightarrow$ $=>$ , $- 6 = 3 B$ $-6=3B$ , $\Rightarrow$ $=>$ , $B = - 2$ $B=-2$

So,

$\frac{2 x^{2} + 4 x + 12}{x^{2} + 7 x + 10} = 2 (1 - (\frac{7}{x + 5} - \frac{2}{x + 2}))$ $(2x^2+4x+12)/(x^2+7x+10)=2(1-(7/(x+5)-2/(x+2)))$

Therefore,

$\int \frac{(2 x^{2} + 4 x + 12) d x}{x^{2} + 7 x + 10} = 2 \int (1 - (\frac{7}{x + 5} - \frac{2}{x + 2})) d x$ $int((2x^2+4x+12)dx)/(x^2+7x+10)=2int(1-(7/(x+5)-2/(x+2)))dx$

$= 2 \int d x - 14 \int \frac{d x}{x + 5} + 4 \int \frac{d x}{x + 2}$ $=2intdx-14intdx/(x+5)+4intdx/(x+2)$

$= 2 x - 14 ln (| x + 5 |) + 4 ln (| x + 2 |) + C$ $=2x-14ln(|x+5|)+4ln(|x+2|)+C$

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How do you integrate $\frac{2 x^{2} + 4 x + 12}{x^{2} + 7 x + 10}$ $(2x^2+4x+12)/(x^2+7x+10)$ using partial fractions?

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Explanation:

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How do you integrate $\frac{2 x^{2} + 4 x + 12}{x^{2} + 7 x + 10}$ $(2x^2+4x+12)/(x^2+7x+10)$ using partial fractions?