TR3.5

Intergration and Anti-Derivation

Houjun Liu 2021-09-27 Mon 12:00

1 Intergration

Antiderivatives table

Function Antidervative
\(x^n\) \(\frac{x^{n+1}}{n+1}+c, x\neq-1\)
\(af(x)\) \(a*(f(x)dx)\)
\(\frac{1}{x}\) \(ln(\ x\ )\)
\(sin(at)\) \(-\frac{cos(t)}{a}\)
\(cos(at)\) \(\frac{sin(t)}{a}\)
\(e^a\) \(e^a\)
\(\frac{1}{1+(ax)^2}\) \(tan^-1(ax)\)
\(\frac{a}{\sqrt{k^2-(ax)^2}}\) \(sin^-1(\frac{ax}{k})\)
\(\frac{-1}{\sqrt{k^2-(ax)^2}}\) \(cos^-1(\frac{ax}{k})\)
\(ln(x)\) \(xln(x)-x\) <= remember this
\(\int f(x)g'(x) dx\) \(f(x)g(x)-\int f'(x)g(x) dx\)
Arc Length of function \(f(x)\) \(\sqrt{1+f'(x)^2} dx\)
Arc length of polar function \(x(t), y(t)\) \(\sqrt{x'(t)^2 + y'(t)^2}(dt)\)
\(r(\theta)\) \(\int_a^B (r(\theta)^2)d\theta\)
\(sec^2(x)\) \(tan(x)\)

Also, fun other things

Function Other Function
\(\cos{2\theta}\) \(1-2sin^2\theta\)
\(\cos{2\theta}\) \(2cos^2\theta-1\)
\(sec^2x-1\) \(tan^2x\)

1.1 Some Limits Too!

\(\lim_{\theta \to \infty} tan^{-1} (\theta) = \frac{\pi}{2}\)

With the reverse product rule, try to make f(x) the simpler derivative, and g(x) the simpler antiderivative

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1.2 Useful thing

  • Intergration by Parts (reverse product rule) (the f(x)g'(x) rule above)
  • u-Substitution (reverse chain rule)
  • Compleeting the Square + arcsin/arctan
  • Long divide, then intergrate