Integration examples
This page shows examples of functions that can be integrated by Symtegration. Each section can also be perused in their own page:
Both sides of each equation was generated directly as LaTeX output with Symtegration, with the integration having been done by Symtegration as well. Integrals that Symtegration cannot derive yet are denoted by \(\bot\).Basic integrals
These are examples of basic integrals integrated by Symtegration.
\(\int 1 \, dx = x\)
\(\int x \, dx = \frac{1}{2} x^{2}\)
\(\int x^{2} \, dx = \frac{x^{3}}{3}\)
\(\int \sqrt{x} \, dx = \frac{2 x^{\frac{3}{2}}}{3}\)
\(\int \frac{1}{x} \, dx = \log x\)
\(\int e^{x} \, dx = e^{x}\)
\(\int \log x \, dx = -x + \left(\log x\right) x\)
\(\int \sin x \, dx = -\cos x\)
\(\int \tan x \, dx = -\log \left\lvert \cos x \right\rvert\)
Integrals with symbols
These are examples of integrals with symbols other than the variable integrated by Symtegration.
\(\int a \, dx = a x\)
\(\int \frac{a}{b} \, dx = \frac{a}{b} x\)
\(\int \mu x \, dx = \frac{1}{2} \mu x^{2}\)
\(\int \sin \left(a x\right) \, dx = -\frac{1}{a} \cos \left(a x\right)\)
\(\int e^{a} \sin x \, dx = -e^{a} \cos x\)
\(\int x \log \left(a x^{2}\right) \, dx = \frac{1}{2 a} \left(-a x^{2} + a x^{2} \log \left(a x^{2}\right)\right)\)
\(\int \left(\log \left(a + x\right) + \log \left(b x\right)\right) \, dx = -\left(a + x\right) + \left(a + x\right) \log \left(a + x\right) + \frac{1}{b} \left(-b x + b x \log \left(b x\right)\right)\)
Integrals of rational functions
These are examples of rational functions integrated by Symtegration. Here, rational functions mean the ratio of two polyomials, not functions of rational numbers.
\(\int \frac{x}{1 + x} \, dx = -\log \left(1 + x\right) + x\)
\(\int \frac{x^{2}}{1 + x^{2}} \, dx = \tan^{-1} \left(-x\right) + x\)
\(\int \frac{x^{2} + x}{x - 1} \, dx = 2 \log \left(-1 + x\right) + 2 x + \frac{1}{2} x^{2}\)
\(\int \frac{1}{x^{3} - x^{5}} \, dx = -\frac{1}{2} \log \left(\frac{3}{2} - \frac{3}{2} x^{2}\right) - \frac{1}{2 x^{2}} + \log \left(9 x\right)\)
\(\int \frac{x^{6} + x + 1}{x^{2} - 1} \, dx = -\frac{1}{2} \log \left(2 + 2 x\right) + x + \frac{1}{3} x^{3} + \frac{1}{5} x^{5} + \frac{3}{2} \log \left(2 - 2 x\right)\)
\(\int \frac{x^{4} - 3 x^{2} + 6}{x^{6} - 5 x^{4} + 5 x^{2} + 4} \, dx = \tan^{-1} x + \tan^{-1} x^{3} + \tan^{-1} \frac{x - 3 x^{3} + x^{5}}{2}\)
Unsupported integrals
Some rational functions cannot be symbolically integrated yet. Some may not be feasible to derive if they require solutions to polynomials of degree 5 or more. Others would be feasible, but would require support for deriving real roots of simultaneous polynomials beyond those that are rational numbers, or support for solving general quartic polynomials.
\(\int \frac{1}{1 + x^{4}} \, dx = \bot\)
\(\int \frac{x}{1 + x^{10}} \, dx = \bot\)
\(\int \frac{x^{5} + 4 x^{2}}{x^{3} + 2 x^{2} + 3} \, dx = \bot\)