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diff --git a/math109/11-15-22.md b/math109/11-15-22.md new file mode 100644 index 0000000..6b55e36 --- /dev/null +++ b/math109/11-15-22.md @@ -0,0 +1,20 @@ +Anti-derivatives (integration): + A function F(x) is called an anti-derivative of f(x) on some interval I if the mathematical condition is that the derivative of F(x) with respect to x is equal to f(x) for all X on I. + +Example: + Find the anti-derivative of 2x - [x^2]. If f(x) = 2x, then F(x) = x^2, +as the derivative of x^2 is 2x. However, x^2 + 1 also derives to 2x. Therefore, F(x) can be equal to x^2 + c. This is "the most general anti-derivative of the function f" + +Trig examples: +1. f(x) = 1/x, F(x) = ln(x) + C +2. g(x) = sin(kx), G(x) = -cos(kx) / k + C +3. h(x) = cos(kx), H(x) = sin(kx) / k + C +4. u(x) = sec^2(kx), U(x) = tan(kx) / k + C + +Exponential examples: +1. f(x) = e^kx, F(x) = e^xk / k + C +2. g(x) = 5^kx, G(x) = 5^kx / ln(5) + C +3. h(x) = 1/(1-x^2), H(x) = arcsin(x) + C + +For any polynomial x^n, the anti-derivative is (x^n+1) / (n + 1) + C, as long as N != -1 + |