To integrate constant times function {constant times function integration}, take integral of k * u(x) * dx = k * (integral of u(x) * dx), where k is constant, and u(x) is function.
Integral from a to b of f(x) * dx equals f(mean) * (b - a) {linear function, integration}. Integral of sum equals sum of integrals.
To integrate power function {power function, integration}: Increase exponent by one and divide by new exponent. Integral of x^p * dx = x^(p + 1) / (p + 1), if p != -1 (p <> -1), so integral of x^3 = x^4 / 4. Integral from x = 1 to x = b of (1/x) * dx equals ln(b).
Integral of e^x = e^x {exponential function, integration}. Integral of b^x = (1 / ln(b)) * b^x.
Integral of ln(e^x) = integral of x {logarithmic function, integral}.
For degree 1, 2, or 3 polynomials P(x), definite integral over interval (a,b) is ((b - a) / 6) * (P(a) + P(a + b) / 2 + P(b)) {midpoint rule}.
Integral of sin(x) = -cos(x) {trigonometric function, integration}. Integral of cos(x) = sin(x). Integral of tan(x) = - ln(|cos(x)|). Integral of cot(x) = ln(|sin(x)|). Integral of sec(x) = ln(|sec(x) + tan(x)|). Integral of csc(x) = ln(|csc(x) - cot(x)|). Integral of (sin(x))^2 = (x - sin(x) * cos(x)) / 2. Integral of (cos(x))^2 = (x + sin(x) * cos(x)) / 2. Integral of (tan(x))^2 = tan(x) - x. Integral of (cot(x))^2 = -cot(x) - x. Integral of (sec(x))^2 = tan(x). Integral of (csc(x))^2 = -cot(x).
If numerator polynomial has higher degree, divide polynomials to get quotient and remainder. Then integrate quotient, integrate remainder, and add results {ratios of polynomials integration}.
To integrate sum of functions {sum of functions integration}: Integral of |u(x) + v(x)| * dx = integral of u(x) * dx + integral of v(x) * dx. Integral of |u(x) - v(x)| * dx = integral of u(x) * dx - integral of v(x) * dx.
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Date Modified: 2022.0225