Rational Fractions

Integrals with rational fractions $\frac{Q(x)}{R(x)}$.

$$\circ\,\,\frac{Q(x)}{R(x)}\longrightarrow P(x)+\frac{K(x)}{R(x)}:\underbrace{\int \frac{Q(x)}{R(x)}=\int P(x)dx+\int \frac{K(x)}{R(x)}dx}_{R(x)-\text{polynomial},\,P(x)-\text{integer part},\,K(x)-\text{fractional part}}$$ $$\circ\,\,R(x)=(x-a_1)^{t_1}\dots(x-a_{n})^{t_n} \underbrace{(x^2+a_1x+b_1)^{k_1}\dots(x^2+a_{m}x+b_{m})^{k_m}}_{\text{irreducible polynomials}}\,\,\,\,\,t, k\,\,\in N:$$ $$\frac{K(x)}{R(x)}=\Bigr[\frac{A_{1}^1}{x-a_1}+\dots +\frac{A_{1}^{t_1}}{(x-a_1)^{t_1}}\Bigr]+\dots+\Bigr[\frac{A_{n}^1}{x-a_n}+\dots +\frac{A_{n}^{t_n}}{(x-a_n)^{t_n}}\Bigr]+$$ $$+\Bigr[\frac{B_1^1x+C_1^1}{x^2+a_1x+b_1}+\dots+\frac{B_1^{k_{1}}x+C_1^{k_1}}{(x^2+a_1x+b_1)^{k_1}}\Bigr]+\dots+\Bigr[\frac{B_m^1x+C_m^1}{x^2+a_mx+b_m}+\dots+\frac{B_m^{k_{m}}x+C_m^{k_m}}{(x^2+a_mx+b_m)^{k_m}}\Bigr]\,_{\text{cofficients method}\,(\text{below})}^{*\text{expansion using undefined}}$$ $$1.\,\int\frac{A}{(x-a)^t}dx=-\frac{A}{(t-1)(x-a)^{t-1}};$$ $$2.\,\int\frac{Bx+C}{(x^2+bx+c)^k}dx=\int\frac{\frac{B}{2}(2x+b)-\frac{Bb}{2}+C}{(x^2+bx+c)^k}dx=-\frac{B}{2(k-1)}\frac{1}{(x^2+bx+c)^{k-1}}\,+\,\Bigr(C-\frac{Bb}{2}\Bigr)\int\frac{dx}{(x^2+bx+c)^k};$$ $$3.\,\int\frac{dx}{x^2+bx+c}=\int\frac{dx}{\Big((x+\frac{b}{2})^2+c-\frac{b^2}{4}\Big)^k}\,\,\,\,_{\text{or with recurrent relation}\,(\text{below})}^{\text{- solve with substitution of } u=\arctan{(x+\frac{b}{2})}}\,\,\,(3)$$

$$\circ\,\,\text{Undefined cofficients method}$$ $$\int\frac{x^2+1}{(x+5)(x^2+x+1)}dx:$$ $$\frac{x^2+1}{(x+5)(x^2+x+1)}=\frac{A}{x+5}+\frac{Bx+C}{x^2+x+1}=\frac{(A+B)x^2+(A+5B+C)x+5C+A}{(x+5)(x^2+x+1)}\Rightarrow\,\,\begin{array}{|c} A = \frac{26}{21} \\ B= -\frac{5}{21} \\ C = -\frac{1}{21} \end{array}$$ $$I=\frac{26}{21}\int\frac{dx}{x+5}-\frac{1}{21}\int\frac{5x+1}{x^2+x+1}dx=\frac{26}{21}\ln{(x+5)}-\frac{5}{42\ln{(x^2+x+1)}}+\frac{\sqrt3}{21}\arctan{\frac{2x+1}{\sqrt3}}$$ $$\circ\,\,\text{Recurrent relation for }(3)$$ $$I_k=\int\frac{dx}{(x^2+bx+c)^k}=\int\frac{dt}{(t^2+p^2)^k}\,\,\,\,\,\large{^{t=x+\frac{b}{2},}_{p^2=c-\frac{b^2}{4}}}$$ $$I_k=\int\frac{dt}{(t^2+p^2)^k}=\frac{1}{p^2}\int\frac{p^2+t^2-t^2}{(t^2+p^2)^k}dt=\frac{1}{p^2}\int\frac{dt}{(t^2+p^2)^{k-1}}-\frac{1}{p^2}\int t^2\frac{dt}{(t^2+p^2)^k}=\frac{1}{p^2}I_{k-1}-\frac{1}{2p^2}\int t\,\frac{d(t^2+p^2)}{(t^2+p^2)^k}=$$ $$=\frac{1}{p^2}I_{k-1}+\frac{t}{2p^2(k-1)(t^2+p^2)^{k-1}}-\frac{1}{2p^2(k-1)}\int\frac{dt}{(t^2+p^2)^{k-1}}=\frac{2k-3}{2p^2(k-1)}I_{k-1}+\frac{t}{2p^2(k-1)(t^2+p^2)^{k-1}}$$

$$I_k=\frac{2k-3}{2p^2(k-1)}I_{k-1}+\frac{t}{2p^2(k-1)(t^2+p^2)^{k-1}}$$

$$\circ\,\,\text{Ostrogradsky's method}$$ $$\int\frac{Q(x)}{R(x)}dx=\frac{Q_1(x)}{R_1(x)}+\int\frac{Q_2(x)}{R_2(x)}dx$$

*coefficients in polynomials $Q_1(x)$, $Q_2(x)$ are found by differentiating both parts and equating it $R_2(x)$ - polynomial containing every root of $R(x)$ each with multiplicity 1, $R_1(x)=\frac{R(x)}{R_2(x)}$, $Q_1(x)$ and $Q_2(x)$ - polynomials with degree 1 less than $R_1(x)$ and $R_2(x)$ respectively.

$$\int\frac{x^3}{(x-1)^{100}}dx$$

$$\int\frac{x^{2n-1}}{x^n+1}dx$$

$$\int\frac{x^{3n-1}}{(x^{2n}+1)^2}dx$$

$$\int\frac{2x+3}{(x-2)(x+5)}dx$$

$$\int\frac{dx}{x(x+1)(x+2)}$$

$$\int\frac{xdx}{(x+1)(x+2)(x+3)}$$

$$\int\frac{dx}{(x^2-1)^2}$$

$$\int\frac{dx}{x^2(1+x^2)^2}$$

$$\int\frac{dx}{(x^2+1)(x^2+2)}$$

$$\int\frac{dx}{x^4-1}$$

$$\int\frac{x^2}{1-x^4}dx$$

$$\int\frac{dx}{x^3+1}$$

$$\int\frac{xdx}{x^3-1}$$

$$\int\frac{dx}{x^4+1}$$

$$\int\frac{x^2}{x^4+1}dx$$

$$\int\frac{x-1}{(x^2+x+1)^2}dx$$

$$\int\frac{dx}{x^4(x-2)^3}$$

$$\int\frac{dx}{(x^2-3x+2)^2}$$

$$\int\frac{2x^2-x^5}{1+x^6}dx$$

$$\int\frac{dx}{x^4+x^2+1}$$

$$\int\frac{dx}{x(1+x^3)^2}$$

$$\int\frac{dx}{x(x^{10}+2)}$$

$$\int\frac{dx}{x(x^{10}+1)^2}$$

$$\int\frac{x^2-1}{x^4+x^2+1}dx$$

$$\int\frac{x^2+1}{x^4+5x^2+1}dx$$

$$\int\frac{dx}{x^6+1}$$

$$\int\frac{x^2-1}{x^4-x^2+1}dx$$

$$\int\frac{x^2+x}{x^6+1}dx$$

$$\int\frac{x^4+1}{x^6+1}dx$$

$$\int\frac{dx}{x^4(x^6+1)}$$

$$\int\frac{xdx}{x^8-1}$$

$$\int\frac{x^7dx}{(1+x^4)^2}$$

$$\int\frac{x^{10}dx}{x^2+x-2}$$

$$\int\frac{x^{11}dx}{(1+x^6)^3}$$

$$\int\frac{1-x^7}{x(1+x^7)}dx$$

$$\int\frac{x^3+1}{x^3-5x^2+6x}dx$$

$$\int\frac{x^4}{x^4+5x^2+4}dx$$

$$\int\frac{xdx}{x^3-3x+2}$$

$$\int\frac{x^2+1}{(x+1)^2(x-1)}dx$$

$$\int\Big(\frac{x}{x^2-3x+2}\Big)^2dx$$

$$\int\frac{x^4-1}{x(x^4-5)(x^5-5x+1)}dx$$

$$\int\frac{dx}{(x+1)(x+2)^2(x+3)^3}$$

$$\int\frac{dx}{x^5+x^4-2x^3-2x^2+x+1}$$

$$\int\frac{x^2+5x+4}{x^4+5x^2+4}dx$$

$$\int\frac{dx}{(x+1)(x^2+1)}$$

$$\int\frac{dx}{(x^2-4x+4)(x^2-4x+5)}$$

$$\int\frac{xdx}{(x-1)^2(x^2+2x+2)}$$

$$\int\frac{dx}{x(x+1)(x^2+x+1)}$$

$$\int\frac{dx}{(x+1)(x^2+1)(x^3+1)}$$

$$\int\frac{dx}{x^5-x^4+x^3-x^2+x-1}$$

$$\int\frac{x^2dx}{x^4+3x^3+\frac{9}{2}x^2+3x+1}$$

$$\int\frac{xdx}{(x-1)^2(x+1)^3}$$

$$\int\frac{dx}{(x^3+1)^2}$$

$$\int\frac{dx}{(x^2+1)^3}$$

$$\int\frac{x^2dx}{(x^2+2x+2)^2}$$

$$\int\frac{dx}{(x^4+1)^2}$$

$$\int\frac{x^2+3x-2}{(x-1)(x^2+x+1)^2}dx$$

$$\int\frac{dx}{(x^4-1)^3}$$

$$\int\frac{x^2+1}{(x^4+x^2+1)^2}dx$$

$$\int\frac{dx}{(x^3+x+1)^3}$$

$$\int\frac{4x^5-1}{(x^5+x+1)^2}dx$$

$$\int\frac{dx}{x^4+2x^3+3x^2+2x+1}$$