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Question Number 142149 by cesarL last updated on 27/May/21

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

7 . \int \frac{4 d x}{x^{2} + 2 x + 4}

8 . \int \frac{3 - 2 x d x}{x^{2} - 64}

9 . \int \frac{3 x - 1}{x^{3} + 5 x^{2} + 6 x} d x

10 . \int \frac{4 - 3 x}{x^{3} - 2 x} d x

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

Answered by mathmax by abdo last updated on 27/May/21

1) I = \int \frac{dx}{x^{2} - 3 x + 2}

x^{2} - 3 x + 2 = 0 \to Δ = 9 - 8 = 1 \Rightarrow x_{1} = \frac{3 + 1}{2} = 2 and x_{2} = \frac{3 - 1}{2} = 1 \Rightarrow

I = \int \frac{dx}{(x - 2) (x - 1)} = \int (\frac{1}{x - 2} - \frac{1}{x - 1}) dx = \log ∣ \frac{x - 2}{x - 1} ∣ + C

Answered by mathmax by abdo last updated on 27/May/21

2) J = \int \frac{4 dx}{x^{2} + 2 x + 4} \Rightarrow J = \int \frac{4 dx}{{(x + 1)}^{2} + 3}

=_{x + 1 = \sqrt{3} y} \int \frac{4 \sqrt{3} dy}{3 (y^{2} + 1)} = \frac{4}{\sqrt{3}} arctany + C

= \frac{4}{\sqrt{3}} \arctan (\frac{x + 1}{\sqrt{3}}) + C

Answered by mathmax by abdo last updated on 27/May/21

3) I = \int \frac{- 2 x + 3}{x^{2} - 64} dx

F (x) = \frac{- 2 x + 3}{x^{2} - 64} = \frac{- 2 x + 3}{(x - 8) (x + 8)} = \frac{a}{x - 8} + \frac{b}{x + 8}

a = \frac{- 16 + 3}{16} = - \frac{13}{16}

b = \frac{16 + 3}{- 16} = - \frac{19}{16} \Rightarrow F (x) = - \frac{13}{16 (x - 8)} - \frac{19}{16 (x + 8)} \Rightarrow

I = - \frac{13}{16} \log ∣ x - 8 ∣ - \frac{19}{16} \log ∣ x + 8 ∣ + C

Answered by mathmax by abdo last updated on 27/May/21

5) J = \int \frac{3 x - 1}{x^{3} + {5 x}^{2} + 6 x} dx

F (x) = \frac{3 x - 1}{x (x^{2} + 5 x + 6)}

x^{2} + 5 x + 6 = 0 \to Δ = 25 - 24 = 1 \Rightarrow x_{1} = \frac{- 5 + 1}{2} = - 2

x_{2} = \frac{- 5 - 1}{2} = - 3 \Rightarrow F (x) = \frac{3 x - 1}{x (x + 2) (x + 3)} = \frac{a}{x} + \frac{b}{x + 2} + \frac{c}{x + 3}

a = - \frac{1}{6}, b = \frac{- 7}{(- 2) (1)} = \frac{7}{2}, c = \frac{- 10}{(- 3) (- 1)} = - \frac{10}{3} \Rightarrow

F (x) = - \frac{1}{6 x} + \frac{7}{2 (x + 2)} - \frac{10}{3 (x + 3)} \Rightarrow J = \int F (x) dx

= - \frac{1}{6} \log ∣ x ∣ + \frac{7}{2} \log ∣ x + 2 ∣ - \frac{10}{3} \log ∣ x + 3 ∣ + C

Answered by Panacea last updated on 27/May/21

Q 11 .

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

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

l n \frac{1}{x} + \frac{1}{2} l n (x^{2} - 1)

l n \frac{(\sqrt{x^{2} - 1})}{x}

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