Całka funkcji wymiernej

[frytusia720]

Oblicz całkę funkcji wymiernej:

\(\int_{}^{} \frac{x ^{2}+2 }{x+2} \mbox{d}x\)
\(\int_{}^{} \frac{ \left(-2x ^{2} +x+3 \right) }{x-1} \mbox{d}x\)
\(\int_{}^{} \frac{ \mbox{d}x }{x \left( x+3\right) }\)
\(\int_{}^{} \frac{- \mbox{d}x }{ \left( x-1\right) \left(x+3 \right) }\)
\(\int_{}^{} \frac{2x-4}{x ^{3}-2x ^{2} } \mbox{d}x\)

[irena]

a)
\(\int\frac{x^2+2}{x+2}dx=\int\frac{x^2-4+6}{x+2}dx=\int(x-2+\frac{6}{x+2})dx=\frac{1}{2}x^2-2x+6ln|x+2|+C\)

b)
\(\int\frac{-2x^2+x+9}{x-1}dx=\int(-2x-1+\frac{8}{x-1)dx=-x^2-x+8ln|x-1|+C\)

c)
\(\int\frac{dx}{x(x+3)}=\\\(\frac{1}{x(x+3)}=\frac{\frac{1}{3}}{x}-\frac{\frac{1}{3}}{x+3}\)\\=\int(\frac{\frac{1}{3}}{x}-\frac{\frac{1}{3}}{x+3})dx=\frac{1}{3}ln|x|-\frac{1}{3}ln|x+3|+C\)

[irena]

d)
\(\int\frac{-dx}{(x-1)(x+3)}=\\\(\frac{-1}{(x-1)(x+3)}=\frac{-\frac{1}{4}}{x-1}+\frac{\frac{1}{4}}{x+3}\)\\=-\frac{1}{4}\int\frac{dx}{x-1}+\frac{1}{4}\int\frac{dx}{x+3}=-\frac{1}{4}ln|x-1|+\frac{1}{4}ln|x+3|+C\)

[irena]

e)
\(\int\frac{2x-4}{x^3-2x^2}dx=\int\frac{2(x-2)}{x^2(x-2)}dx=\int\frac{2}{x^2}dx=-\frac{2}{x}+C\)