x-2-x-2-5-2-x-3-2-dx-

Question Number 88307 by M±th+et£s last updated on 09/Apr/20

\int \frac{x^{2}}{x^{2} - \frac{5}{2} x - \frac{3}{2}} d x

Answered by TANMAY PANACEA. last updated on 09/Apr/20

\int \frac{2 x^{2}}{2 x^{2} - 5 x - 3}

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

2 \int \frac{x^{2}}{2 x (x - 3) + 1 (x - 3)} d x

\int \frac{2 x^{2} d x}{(x - 3) (2 x + 1)}

\frac{2 x^{2}}{(x - 3) (2 x + 1)} = 1 + \frac{a}{x - 3} + \frac{b}{2 x + 1}

2 x^{2} = 2 x^{2} - 5 x - 3 + 2 a x + a + b x - b

0 = x (2 a + b - 5) + (a - b - 3)

2 a + b - 5 = 0

a - b - 3 = 0

3 a = 8 \to a = \frac{8}{3}

b = \frac{8}{3} - 3 = \frac{- 1}{3}

\int d x + \frac{8}{3} \int \frac{d x}{x - 3} + \frac{- 1}{3} \int \frac{d x}{2 x + 1}

\int d x + \frac{8}{3} \int \frac{d x}{x - 3} + \frac{- 1}{6} \int \frac{d x}{x + \frac{1}{2}}

x + \frac{8}{3} l n (x - 3) + \frac{- 1}{6} l n (x + \frac{1}{2}) + c

Commented by M±th+et£s last updated on 09/Apr/20

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