Integration of Irrational Functions

If the integrand contains a root of some degree of a linear-fractional function $\frac{\alpha x+\beta }{\gamma x+\delta }$, then this irrationality can be removed by the substitution $\frac{\alpha x+\beta }{\gamma x+\delta }=t^{n}$, where $n$ is a suitably chosen exponent. Using this technique, find the integral $\int \frac{dx}{\sqrt{2x-1}-\sqrt[3]{2x-1}}$.

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

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

Find the integral $\int \frac{x\, dx}{\sqrt{x+1}-\sqrt[3]{x+1}}$.

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

Find the integral $\int \frac{2+\sqrt[3]{x}}{\sqrt[6]{x}+\sqrt[3]{x}+\sqrt{x}+1}\, dx$.

The following four integrals, although of a different type, are also found by a substitution of the form $\frac{x-a}{x-b}=t^{n}$. Find the integral $\int \frac{dx}{\sqrt{(x-1)^{3}(x-2)}}$.

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

Find the integral $\int \frac{dx}{\sqrt[3]{(x+1)^{2}(x-1)^{4}}}$.

Find the integral $\int \frac{dx}{\sqrt[3]{(x-1)^{7}(x+1)^{2}}}$.

The following three integrals are found by similar techniques. Find the integral $\int \frac{\sqrt[3]{1+\sqrt[4]{x}}}{\sqrt{x}}\, dx$.

Find the integral $\int \sqrt[3]{\frac{(x+1)^{5}}{(x-1)^{2}}}\, dx$.

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

The following integrals have the form $\int x^{m}(ax^{n}+b)^{p}\, dx$, where $m$, $n$, and $p$ are rational numbers. They reduce to integrals of rational functions in three cases: 1) if $p$ is an integer, even if negative (substitution $x=t^{N}$, where $N$ is the common denominator of the fractions $m$ and $n$); 2) if $\frac{m+1}{n}$ is an integer (substitution $ax^{n}+b=t^{M}$, where $M$ is the denominator of $p$); 3) if $\frac{m+1}{n}+p$ is an integer (substitution $a+bx^{-n}=t^{M}$, where $M$ is the denominator of $p$). If none of the three numbers $p$, $\frac{m+1}{n}$, and $\frac{m+1}{n}+p$ is an integer, then by Chebyshev's theorem integrals of this form cannot be expressed by a finite combination of elementary functions.

Compute the integral $\int \frac{dx}{\sqrt{x}(\sqrt[3]{x}+1)^{2}}$.

Compute the integral $\int \frac{dx}{\sqrt[4]{x^{2}}(\sqrt[3]{x}+1)^{3}}$.

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

Find the integral $\int \frac{x^{7}\, dx}{\sqrt{x^{2}-1}}$.

Find the integral $\int \frac{x^{5}\, dx}{\sqrt[3]{(x^{2}-1)^{2}}}$.

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

Find the integral $\int \frac{dx}{x\sqrt[3]{1+x^{5}}}$.

Find the integral $\int x^{7}\sqrt{1+x^{2}}\, dx$.

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

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

Find the integral $\int \sqrt[3]{x-x^{3}}\, dx$.

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

The following integrals are solved by a technique similar to that used in the third case of the previous type. Find the integral $\int \frac{dx}{(5x^{3}+3)\sqrt[3]{4x^{3}+3}}$.

Find the integral $\int \frac{dx}{(x^{4}+3)\sqrt[3]{2x^{4}+1}}$.

In many cases, when finding integrals of differential binomials, i.e., integrals of the form $u_{m,p}=\int x^{m}(ax^{n}+b)^{p}\, dx$, it is useful first to reduce the integral to another one in which the quantities $m$ and $p$ have more convenient values. This can be done using reduction formulas. The main ones have the following form:

Using the reduction formulas, find the integral $\int \frac{x^{6}\, dx}{\sqrt{x^{2}+1}}$.

Find the integral $\int \frac{x^{8}\, dx}{\sqrt{x^{2}-1}}$.

Find the integral $\int \frac{dx}{x^{7}\sqrt{x^{4}+1}}$.

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

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

Find the integral $\int \frac{x^{9}\, dx}{\sqrt[3]{x^{3}+1}}$.

A large number of the following problems give integrals containing the square root of a quadratic polynomial. All of them can be reduced to integrals of rational functions by one of the three Euler substitutions:

where $x_{1}$ is a root of the polynomial $ax^{2}+bx+c$.

However, in most cases the Euler substitutions lead to longer computations than direct methods. To apply them one should become familiar with three basic types of the integrals under study.

I. Integrals of the form $\int \frac{f(x)\, dx}{\sqrt{ax^{2}+bx+c}}$, where $f(x)$ is a polynomial. In most cases the method of undetermined coefficients is convenient. Here we write the presumed identity

Differentiating both sides of the equality, we obtain

Clearing the denominator, we obtain polynomials of degree $n$ on both sides of the equality. Equating coefficients of like powers of $x$, we obtain equations from which the coefficients $A_{1}, A_{2}, \ldots , A_{n}$ are found one after another.

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

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

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

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

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

Find the integral $\int x^{2}\sqrt{x^{2}+2}\, dx$.

The preceding problems contained integrals of the form $\int \frac{f(x)\, dx}{\sqrt{ax^{2}+bx+c}}$, where $f(x)$ is a polynomial. In order to be able to integrate the most general integral of the form $\int f(x,y)\, dx$, where $f(x,y)$ is a rational function of $x$ and $y$, and $y=\sqrt{ax^{2}+bx+c}$, it is important to be able to find integrals of two further types, which we denote by the numerals II and III.

II. Integrals of the form $\int \frac{f_{m}(x)\, dx}{(x-\alpha )^{n}\sqrt{ax^{2}+bx+c}}$, where $f_{m}(x)$ is a polynomial of degree $m$ less than $n$. These integrals reduce to integrals of the previous class by the substitution $x-\alpha =\frac{1}{t}$. If the degree of the polynomial $f_{m}(x)$ is not less than $n$, then the integer part of the fraction $\frac{f_{m}(x)}{(x-\alpha )^{n}}$ should be extracted first.

Using these methods, find the integral $\int \frac{dx}{x\sqrt{x^{2}-1}}$.

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

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

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

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

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

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

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

III. The third and most complicated type is represented by integrals

where the roots of the trinomial $x^{2}+px+q$ are imaginary. If $p=b=0$, then

Here the first integral is found by the substitution $ax^{2}+c=u^{2}$, and the second by the substitution $a+cx^{-2}=v^{2}$. In the general case the integral $I$ is reduced to this simplest case by the substitution $x=\frac{\alpha t+\beta }{t+1}$. This gives

where $f_{n}(t)$ is a polynomial of degree $n$, and the coefficients $B$ and $B_{1}$ have the following values:

Setting $B=0$, $B_{1}=0$, we obtain equations giving real values for $\alpha$ and $\beta$ for real $p, q, a, b, c$, provided at least one of the polynomials has imaginary roots. With this choice of the numbers $\alpha$ and $\beta$, the integral $I$ splits into a sum of integrals of two types:

which are integrated respectively by the substitutions:

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

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

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

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

Find the integral $\int \frac{(2x+1)\, dx}{(3x^{2}+4x+4)\sqrt{x^{2}+6x-1}}$.

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

The most general integral of the form $\int f(x,\sqrt{ax^{2}+bx+c})\, dx$, where $f(x,y)$ is a rational function of $x$ and $y$, reduces to integrals of the three previous types. For this it suffices, by rationalizing the denominator, to transform the function $f(x,y)$ to the form $\varphi (x)+\psi (x)y$, where $\varphi (x)$ and $\psi (x)$ are rational functions of $x$. Here $\psi (x)y=\frac{\psi (x)y^{2}}{y}=\frac{\psi (x)(ax^{2}+bx+c)}{y}=\frac{\omega (x)}{\sqrt{ax^{2}+bx+c}}$. Thus the matter reduces to finding the integral of the rational function $\varphi (x)$ and of $\frac{\omega (x)}{\sqrt{ax^{2}+bx+c}}$, where $\omega (x)$ is a rational fraction. In the latter integral the rational fraction $\omega (x)$ can be transformed by extracting the integer part and decomposing into partial fractions. This yields integrals of the three types already considered. In simple cases part of these transformations is unnecessary.

Find the integral $\int \sqrt{\frac{a-x}{x-b}}\, dx$.

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

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

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

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

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

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

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

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

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

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

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

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

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

The substitution $\frac{2ax+b}{\sqrt{ax^{2}+bx+c}}=u$ is sometimes called the Abel substitution; it is convenient for finding the integral of the function $(ax^{2}+bx+c)^{-n-\frac{1}{2}}$, where $n$ is an integer.

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

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

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

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

Integrals $\int f(x,y)\, dx$, where $f(x,y)$ is a rational function and $y=\sqrt{Ax^{4}+Bx^{3}+Cx^{2}+Dx+E}$, called elliptic integrals, generally cannot be integrated in closed form unless there are special relations among the coefficients of the function $f(x,y)$ or of the polynomial $Ax^{4}+Bx^{3}+Cx^{2}+Dx+E$.

In the following examples with palindromic polynomials, one of the substitutions $x+\frac{1}{x}=u$ or $x-\frac{1}{x}=v$ makes it comparatively easy to obtain the integral. Find the integral $\int \frac{x^{2}-1}{x^{2}+1}\frac{dx}{\sqrt{x^{4}+1}}$.

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

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

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

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

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

Theorem (Abel). Let $y$ be an algebraic function of $x$ of degree $n$, i.e., $y$ satisfies the equation $\varphi (x,y)=0$, where $\varphi (x,y)$ is a polynomial in $x$ and $y$ of degree $n$ in $y$. If the integral $\int f(x,y)\, dx$ of the function $f(x,y)$, rational in $x$ and $y$, can be expressed through algebraic functions, then the equality

must hold, where $p_{0}(x), p_{1}(x), \ldots , p_{n-1}(x)$ are rational functions of $x$.

Corollary. If $y=\sqrt[m]{\varphi (x)}$, where $\varphi (x)$ is a polynomial of degree $n$, and if the integral $\int \frac{a_{0}x^{\lambda }+a_{1}x^{\lambda -1}+\ldots +a_{\lambda }}{y^{k}}\, dx$ is expressed by an algebraic function, then it must equal $f(x)y^{m-k}$, where $f(x)$ is a polynomial of degree $\lambda =n+1$.

Find the following algebraic integrals. Find the integral $\int \frac{3x^{4}+2x^{3}+8x^{2}+6x+2}{\sqrt[3]{(x^{3}+3x+1)^{2}}}\, dx$.

Find the integral $\int \frac{5x^{5}+11x^{3}+3x^{2}}{\sqrt[3]{x^{3}+3x+1}}\, dx$.

Find the integral $\int \frac{5x^{7}+17x^{4}+4x^{3}}{\sqrt[4]{(x^{4}+4x+1)^{3}}}\, dx$.