List of integrals of rational functions
- List of integrals of rational functions
The following is a list of integrals (antiderivative functions) of rational functions. For a more complete list of integrals, see lists of integrals.:
:
:
: intfrac{1}{(ax^2+bx+c)^n} dx= frac{2ax+b}{(n-1)(4ac-b^2)(ax^2+bx+c)^{n-1+frac{(2n-3)2a}{(n-1)(4ac-b^2)}intfrac{1}{(ax^2+bx+c)^{n-1 dx,!: intfrac{x}{(ax^2+bx+c)^n} dx= frac{bx+2c}{(n-1)(4ac-b^2)(ax^2+bx+c)^{n-1-frac{b(2n-3)}{(n-1)(4ac-b^2)}intfrac{1}{(ax^2+bx+c)^{n-1 dx,!: intfrac{1}{x(ax^2+bx+c)} dx= frac{1}{2c}lnleft|frac{x^2}{ax^2+bx+c}
ight|-frac{b}{2c}intfrac{1}{ax^2+bx+c} dx
: int frac{dx}{x^{2^n} + 1} = sum_{k=1}^{2^{n-1 left { frac{1}{2^{n-1 left [ sin(frac{(2k -1) pi}{2^n}) arctan [left(x - cos(frac{(2k -1) pi}{2^n})
ight ) csc(frac{(2k -1) pi}{2^n}) ]
ight] - frac{1}{2^n} left [ cos(frac{(2k -1) pi}{2^n}) ln left | x^2 - 2 x cos(frac{(2k -1) pi}{2^n}) + 1
ight |
ight ]
ight }
Any rational function can be integrated using above equations and partial fractions in integration, by decomposing the rational function into a sum of functions of the form:: frac{ex + f}{left(ax^2+bx+c
ight)^n}.
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