całki

[anilina]

1. \(\int_{}^{} \frac{x^4}{x^2+1} \cdot dx\)

2. \(\int_{}^{} \frac{dx}{x^3+x^2+x}\)

3.\(\int_{}^{} \frac{dx}{x^4+x^2+1}\)

[domino21]

\(\int \frac{x^4}{x^2+1} dx =\int \frac{x^4+x^2}{x^2+1} dx -\frac{x^2}{x^2+1}dx=\int x^2 dx -\int \frac{x^2+1}{x^2+1}dx +\int \frac{1}{x^2+1}dx =
=\int x^2 dx -\int dx+\int \frac{dx}{x^2+1} =\frac{1}{3}x^3-x + arc tg x +C\)

[anilina]

dzięki! a dałoby się dwie pozostałe również?

[Pol]

b) \(\frac{1}{x^3+x^2+x} = \frac{1}{x(x^2+x^1+1)} = \frac{A}{x}+\frac{Bx+C}{x^2+x+1}\)

\(1 = Ax^2+Ax+A+Bx^2+Cx
\{A+B=0
A+C=0
A=1\)

\(A = 1, \ B = -1, \ C = -1\)

\(\int_{}^{} \frac{dx}{x^3+x^2+x} = \int \frac 1 x dx + \int \frac{-x-1}{x^2+x+1}dx=ln|x|\ -\ \frac 1 2 \int \frac{2x+2}{x^2+x+1}dx\)

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

dalej już prosto

c) wskazówka: \(x^4+x^2+1 = (x^2+1)^2-x^2 = (x^2+x+1)(x^2-x+1)\)