Three-column integral table

By Stefan Nikolaj on July 28, 2024. Tags: math.

For my university math course, I needed to make a cheat sheet with some common integrals and derivatives. I remember once seeing a table of functions with their integrals and derivatives, but I couldn’t find it again, so I made it again with the functions I found myself using the most. Below is a screenshot of the table (because I couldn’t find an easy way to embed Obsidian’s Markdown-LaTeX mix) and an unrendered LaTeX version. In the table, k and n are constants. For increased readability, you can replace k with 1.

Screenshot:

Unrendered LaTeX version:

DerivativeNormalIntegral
$nx^{n-1}$$x^n$$\frac{x^{n+1}}{n+1}$
$\frac{1}{x}$$ln(kx)$$x(ln(kx)-1)$
$kcos(kx)$$sin(kx)$$-\frac{1}{k}cos(kx)$
$ksec^{2}(kx)$$tan(kx)$$\frac{1}{k}ln(sec(kx))$ or $-\frac{1}{k}ln(cos(kx))$
$-kcsc^{2}(kx)$$cot(kx)$$\frac{1}{k}ln(sin(kx))$ or $-\frac{1}{k}ln(csc(kx))$
$ktan(kx)sec(kx)$$sec(kx)$$\frac{1}{k}ln(sec(kx)+tan(kx))$
$-kcot(kx)csc(kx)$$csc(kx)$$\frac{1}{k}ln(csc(kx)-cot(kx))$ or
$\frac{1}{k}(ln(sin(\frac{kx}{2})) -ln(cos(\frac{kx}{2})))$
$\frac{k}{\sqrt{1-k^{2}x^{2}}}$$arcsin(kx)$$\frac{\sqrt{1-k^{2}x^{2}}}{k}+xarcsin(kx)$
$-\frac{k}{\sqrt{1-k^{2}x^{2}}}$$arccos(kx)$$-\frac{\sqrt{1-k^{2}x^{2}}}{k}+xarccos(kx)$
$\frac{k}{k^{2}x^{2}+1}$$arctan(kx)$$-\frac{ln(k^{2}x^{2}+1)}{2k}+xarctan(kx)$
$-\frac{k}{k^{2}x^{2}+1}$$arccot(kx)$$\frac{ln(k^{2}x^{2}+1)}{2k}+xarccot(kx)$
$\frac{1}{x\sqrt{k^{2}x^{2}-1}}$$arcsec(kx)$$xarcsec(kx)-\frac{arctan(\frac{kx}{\sqrt{k^{2}x^{2}-1}})}{k}$
$-\frac{1}{x\sqrt{k^{2}x^{2}-1}}$$arccsc(kx)$$xarccsc(kx)+\frac{arctan(\frac{kx}{\sqrt{k^{2}x^{2}-1}})}{k}$
$ke^{kx}$$e^{kx}$$\frac{1}{k}e^{kx}$
$kln(n)n^{kx}$$n^{kx}$$\frac{n^{kx}}{kln(n)}$
$\frac{1}{xln(10)}$$log_{10}(kx)$$\frac{x(ln(x)-1)}{ln(10)}$
$\frac{1}{\sqrt{x^{2}+k}}$$ln(x+\sqrt{x^{2}+k})$/
$\frac{1}{x^{2}-a^{2}}$$\frac{1}{2a}ln \frac{x-a}{x+a}$/
$2sin(x)cos(x)$$sin^{2}(x)$$\frac{1}{2}(x-sin(x)cos(x))$ or $\frac{x}{2}-\frac{1}{4}sin(2x)$
$-2sin(x)cos(x)$$cos^{2}(x)$$\frac{1}{2}(x+sin(x)cos(x))$ or $\frac{x}{2}+\frac{1}{4}sin(2x)$

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