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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
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