помогите решить интеграл (x^2-3)dx/(x^4+5x^2+6). Created by daria23258. matematika-ru

(x^4+5x^2+6)...

July 2022 1 4 Report

помогите решить интеграл (x^2-3)dx/(x^4+5x^2+6)

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let5 $\int \frac{x^2-3}{x^4+5x^2+6} \, dx=\int \frac{x^2-3}{(x^2+3)(x^2+2)} \, dx= \\
 \frac{x^2-3}{(x^2+3)(x^2+2)}= \frac{A}{x^2+3} + \frac{B}{x^2+2} = \frac{Ax^2+2A+Bx^2+3B}{(x^2+3)(x^2+2)} = \frac{(A+B)x^2+(2A+3B)}{(x^2+3)(x^2+2)} \\
 \left \{ {{A+B=1} \atop {2A+3B=-3}} \right. \ \left \{ {{A=6} \atop {B=-5}} \right. \\ 
 \frac{x^2-3}{(x^2+3)(x^2+2)}= \frac{6}{x^2+3} - \frac{5}{x^2+2} \\
=6\int \frac{dx}{x^2+3} -5\int \frac{dx}{x^2+2}$ $= \frac{6}{ \sqrt{3} } arctg \frac{x}{ \sqrt{3} } - \frac{5}{ \sqrt{2} } arag \frac{x}{ \sqrt{2} } +C$

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(1+x^2)...

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