How do you integrate $\int \frac{3 x^{3} - 22 x^{2} + 27 x + 46}{x^{2} - 8 x + 15}$ $int (3x^3 - 22x^2 + 27x + 46)/(x^2 - 8x + 15)$ using partial fractions?

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How do you integrate $\int \frac{3 x^{3} - 22 x^{2} + 27 x + 46}{x^{2} - 8 x + 15}$ $int (3x^3 - 22x^2 + 27x + 46)/(x^2 - 8x + 15)$ using partial fractions?

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Answer 1 · 2016-10-02T21:25:49

$\int \frac{3 x^{3} - 22 x^{2} + 27 x + 46}{x^{2} - 8 x + 15} d x = \frac{3}{2} x^{2} + 2 x - 5 ln | x - 3 | + 3 ln | x - 5 | + C$ $int (3x^3-22x^2+27x+46)/(x^2-8x+15) dx = 3/2x^2+2x-5ln abs(x-3)+3ln abs(x-5) + C$

Explanation:

$\frac{3 x^{3} - 22 x^{2} + 27 x + 46}{x^{2} - 8 x + 15} = \frac{(3 x^{3} - 24 x^{2} + 45 x) + (2 x^{2} - 16 x + 30) + (- 2 x + 16)}{x^{2} - 8 x + 15}$ $(3x^3-22x^2+27x+46)/(x^2-8x+15) = ((3x^3-24x^2+45x)+(2x^2-16x+30)+(-2x+16))/(x^2-8x+15)$

$\frac{3 x^{3} - 22 x^{2} + 27 x + 46}{x^{2} - 8 x + 15} = 3 x + 2 + \frac{- 2 x + 16}{x^{2} - 8 x + 15}$ $color(white)((3x^3-22x^2+27x+46)/(x^2-8x+15)) = 3x+2+(-2x+16)/(x^2-8x+15)$

$\frac{3 x^{3} - 22 x^{2} + 27 x + 46}{x^{2} - 8 x + 15} = 3 x + 2 + \frac{- 2 x + 16}{(x - 3) (x - 5)}$ $color(white)((3x^3-22x^2+27x+46)/(x^2-8x+15)) = 3x+2+(-2x+16)/((x-3)(x-5))$

$\frac{3 x^{3} - 22 x^{2} + 27 x + 46}{x^{2} - 8 x + 15} = 3 x + 2 + \frac{A}{x - 3} + \frac{B}{x - 5}$ $color(white)((3x^3-22x^2+27x+46)/(x^2-8x+15)) = 3x+2+A/(x-3)+B/(x-5)$

We can determine $A$ $A$ and $B$ $B$ using Heaviside's cover up method...

$A = \frac{- 2 (3) + 16}{3 - 5} = \frac{10}{- 2} = - 5$ $A = (-2(color(blue)(3))+16)/(color(blue)(3)-5) = 10/(-2) = -5$

$B = \frac{- 2 (5) + 16}{5 - 3} = \frac{6}{2} = 3$ $B = (-2(color(blue)(5))+16)/(color(blue)(5)-3) = 6/2 = 3$

So:

$\int \frac{3 x^{3} - 22 x^{2} + 27 x + 46}{x^{2} - 8 x + 15} d x = \int 3 x + 2 - \frac{5}{x - 3} + \frac{3}{x - 5} d x$ $int (3x^3-22x^2+27x+46)/(x^2-8x+15) dx = int 3x+2-5/(x-3)+3/(x-5) dx$

$\int \frac{3 x^{3} - 22 x^{2} + 27 x + 46}{x^{2} - 8 x + 15} d x = \frac{3}{2} x^{2} + 2 x - 5 ln | x - 3 | + 3 ln | x - 5 | + C$ $color(white)(int (3x^3-22x^2+27x+46)/(x^2-8x+15) dx) = 3/2x^2+2x-5ln abs(x-3)+3ln abs(x-5) + C$

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How do you integrate $\int \frac{3 x^{3} - 22 x^{2} + 27 x + 46}{x^{2} - 8 x + 15}$ $int (3x^3 - 22x^2 + 27x + 46)/(x^2 - 8x + 15)$ using partial fractions?

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How do you integrate int (3x^3 - 22x^2 + 27x + 46)/(x^2 - 8x + 15)∫3x3−22x2+27x+46x2−8x+15int (3x^3 - 22x^2 + 27x + 46)/(x^2 - 8x + 15) using partial fractions?

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How do you integrate $\int \frac{3 x^{3} - 22 x^{2} + 27 x + 46}{x^{2} - 8 x + 15}$ $int (3x^3 - 22x^2 + 27x + 46)/(x^2 - 8x + 15)$ using partial fractions?