Integration of Rational Functions

Find the integral $\int \frac{x\, dx}{x^{2}-3x+2}$, using the identity

Find the integral $\int \frac{2x+3}{x^{2}-5x+6}\, dx$, using the identity

in which the coefficients $A$ and $B$ are easily found by equating coefficients of like powers of $x$ after clearing the denominator.

Find the integral $\int \frac{dx}{x^{4}-1}$.

Find the integral $\int \frac{x\, dx}{x^{3}-1}$.

Find the integral $\int \frac{2x^{2}+4}{x^{3}-x^{2}+x-1}\, dx$.

Find the integral $\int \frac{dx}{x^{2}(x-1)}$.

Find the integral $\int \frac{dx}{(x+a)(x^{2}+b^{2})}$.

Find the integral $\int \frac{x^{2}-x+2}{x^{4}-5x^{2}+4}\, dx$.

Find the integral $\int \frac{x^{2}\, dx}{(x-1)(x-2)(x-3)}$.

Find the integral $\int \frac{x\, dx}{x^{3}-6x^{2}+11x-6}$.

Find the integral $\int \frac{dx}{x^{3}+1}$.

Find the integral $\int \frac{x^{5}+1}{x^{6}+x^{4}}\, dx$.

Find the integral $\int \frac{x^{3}+x^{2}+2}{x(x^{2}-1)^{2}}\, dx$.

Find the integral $\int \frac{x^{3}-2x^{2}+4}{x^{3}(x-2)^{2}}\, dx$.

Find the integral $\int \frac{x^{2}\, dx}{x^{4}-1}$.

Find the integral $\int \frac{x\, dx}{x^{3}+1}$.

Find the integral $\int \frac{9x^{2}-14x+1}{x^{3}-2x^{2}-x+2}\, dx$.

Find the integral $\int \frac{dx}{(x^{2}+9)^{2}}$.

Find the integral $\int \frac{dx}{x^{3}+2x-3}$.

In the following integrals, to factor the denominator it is convenient to add and subtract the term $ax^{2}$, where $a$ is a suitably chosen coefficient. Find the integral $\int \frac{dx}{x^{4}+1}$.

Find the integral $\int \frac{x^{2}\, dx}{x^{4}+x^{2}+1}$.

Find the integral $\int \frac{(x^{2}+2)\, dx}{x^{4}+x^{2}+4}$.

Find the integral $\int \frac{dx}{x^{4}+x^{2}+1}$.

If the denominator factors into simple factors of the first degree, then for decomposing the fraction into partial fractions the formula

is convenient, where $a_{1}, a_{2}, \ldots , a_{n}$ are the roots of the polynomial $\psi (x)$. If among the roots there is a pair of complex conjugates, then in the sum on the right-hand side the two corresponding terms can be replaced by a single quantity:

Here $\bar{a}=\alpha -\beta i$ is the number complex conjugate to $a=\alpha +\beta i$.

Find the integral $\int \frac{6\, dx}{x(x-1)(x-2)(x-3)}$.

Find the integral $\int \frac{24\, dx}{x(x^{2}-1)(x^{2}-4)}$.

Find the integral $\int \frac{dx}{x^{2n}+1}$.

The following integrals are reduced to integrals of the previous types after extracting the integer part of the integrand fraction. Find the integral $\int \frac{x^{4}\, dx}{x^{2}+3}$.

Find the integral $\int \frac{x^{6}\, dx}{x^{2}-1}$.

Find the integral $\int \frac{x^{3}\, dx}{x^{2}+x+1}$.

Find the integral $\int \frac{x^{3}\, dx}{x^{2}-3x+2}$.

Find the integral $\int \frac{x^{6}+16}{x^{4}-4}\, dx$.

Find the integral $\int \frac{x^{4}\, dx}{x^{3}-6x^{2}+11x-6}$.

Find the integral $\int \frac{x^{2}+x+1}{x^{2}-x+1}\, dx$.

Find the integral $\int \frac{x^{4}+1}{x^{3}-x^{2}+x-1}\, dx$.

Find the integral $\int \frac{x^{6}\, dx}{(x^{2}-1)^{2}}$.

Using the identity $4a(ax^{2}+bx+c)-(2ax+b)^{2}=4ac-b^{2}$ and integrating by parts the integral $\int \frac{(2ax+b)^{2}\, dx}{(ax^{2}+bx+c)^{n}}$, one can prove the reduction formula

where $u_{n}=\int \frac{dx}{(ax^{2}+bx+c)^{n}}$.

Find the following integrals using the reduction formula. Find the integral $\int \frac{dx}{(x^{2}+1)^{3}}$.

Find the integral $\int \frac{dx}{(x^{2}+x+1)^{3}}$.

Find the integral $\int \frac{dx}{(x^{2}+4)^{4}}$.

Find the integral $\int \frac{dx}{(x^{2}-3x+3)^{2}}$.

Prove the equality

Prove the equality

Find the integrals. Find the integral $\int \frac{x^{4}\, dx}{(x^{2}+3)^{3}}$.

Find the integral $\int \frac{x^{6}\, dx}{(x^{2}-5)^{4}}$.

Find the integral $\int \frac{x^{6}\, dx}{(x^{2}+4)^{4}}$.

Find the integral $\int \frac{x^{4}\, dx}{(x^{2}-3)^{3}}$.

In the following problems the integrand has the form $x^{m}(ax^{n}+b)^{-p}$. For these the substitution $x^{\sigma }=t$ is useful, where $\sigma$ is the greatest common divisor of $m+1$ and $n$. Find the integral $\int \frac{x^{5}\, dx}{x^{4}+1}$.

Find the integral $\int \frac{x^{3}\, dx}{(x^{8}+4)^{2}}$.

Find the integral $\int \frac{x^{3}\, dx}{x^{8}+1}$.

Find the integral $\int \frac{x^{5}\, dx}{(x^{12}+1)^{2}}$.

Find the integral $\int \frac{dx}{x(x^{3}+4)}$.

Find the integral $\int \frac{x^{5}\, dx}{(x^{2}+2)^{2}}$.

Find the integral $\int \frac{x^{3}+1}{x(x^{3}-8)}\, dx$.

Find the integral $\int \frac{x^{8}-1}{x(x^{8}+1)}\, dx$.

Find the integral $\int \frac{dx}{x(x^{6}+1)^{2}}$.

Find the integral $\int \frac{x^{5}\, dx}{x^{12}-1}$.

In the following problems the integral is found after the denominator is factored. Find the integral $\int \frac{x^{3}+4x^{2}+6x}{x^{4}+2x^{3}+3x^{2}+4x+2}\, dx$.

Find the integral $\int \frac{2x^{2}+x+4}{x^{4}+x^{3}+2x^{2}-x+3}\, dx$.

Find the integral $\int \frac{4x^{3}-x^{2}+10x+5}{x^{4}-x^{3}+4x^{2}+3x+5}\, dx$.

Find the integral $\int \frac{4x^{3}-17x^{2}+28x-15}{x^{4}-7x^{3}+20x^{2}-27x+15}\, dx$.

Find the algebraic part of the integrals. Find the integral $\int \frac{x^{2}+1}{x(x^{3}+1)^{2}}\, dx$.

Find the integral $\int \frac{dx}{x^{2}(x^{2}+1)^{2}}$.

Find the integral $\int \frac{2-3x+x^{2}}{(x+1)^{2}(x^{2}+x+1)^{2}}\, dx$.

Find the integral $\int \frac{4x^{5}-1}{(x^{5}+x+1)^{2}}\, dx$.

A number of the following problems are solved by various special techniques that avoid decomposing the fraction into partial fractions. The linear-fractional substitution $\frac{x-a}{x-b}=t$ conveniently gives the integral $\int \frac{dx}{(x+1)^{2}(x-2)^{3}}$.

Find the integral $\int \frac{dx}{(x^{2}-x)^{3}}$.

Find the integral $\int \frac{dx}{(x^{2}-1)^{4}}$.

Find the integral $\int \frac{dx}{(x-1)^{2}(x^{2}-1)^{3}}$.

The substitutions $x+\frac{1}{x}=u$ and $x-\frac{1}{x}=v$ make it easy to find the following integrals, which contain palindromic polynomials and the factor $x^{2}-1$ or $x^{2}+1$. Find the integral $\int \frac{x^{2}-1}{x^{4}+3x^{3}+5x^{2}+3x+1}\, dx$.

Find the integral $\int \frac{x^{2}-1}{x^{2}+1}\frac{dx}{x^{2}+x+1}$.

Find the integral $\int \frac{x^{2}+1}{x^{4}+2x^{3}+3x^{2}-2x+1}\, dx$.

Find the integral $\int \frac{x^{2}+1}{x^{4}+x^{2}+1}\, dx$.

Find the integral $\int \frac{dx}{x^{4}+x^{2}+1}$, using the identities

Find the integral $\int \frac{x^{4}+1}{x^{6}+1}\, dx$, by subtracting and adding the quantity $x^{2}$ in the numerator.

Find the integral $A=\int \frac{dx}{x^{4}+1}$, introducing the auxiliary integral $B=\int \frac{x^{2}\, dx}{x^{4}+1}$ and noting that the integrals $A+B$ and $A-B$ are easily found after the substitutions $x-\frac{1}{x}=u$ and $x+\frac{1}{x}=v$.

Find the value of the integral $A=\int \frac{dx}{x^{6}+1}$, introducing the integral $B=\int \frac{x^{4}\, dx}{x^{6}+1}$.

Find the integrals $C=\int \frac{x^{2}\, dx}{x^{8}+1}$ and $D=\int \frac{x^{4}\, dx}{x^{8}+1}$.