Calculus indefinite integral of a function f(x) this is search such function g(x) then its derivative equal f(x). In abbreviation: ∫ f(x) dx = g(x) because g'(x) = f(x). Examples: C - any number. ∫ x dx = 1/2 x^2 + C because ((1/2)x^2 + C)' = ((1/2)x^2)' + (C)' = x. ∫ x^2 dx = (1/3)x^3 + C because ((1/3)x^3 + C)' = ((1/3)x^3)' + (C)' = x^2. ∫(cos x)dx = sin x + C because (sin x + C)' = (sin x)' + (C)' = cos x. Basic Integrals. Formulas. Examples. ∫ x^a dx = x^{a+1} / a+1 + C. ∫ x^2 dx = x^3/3 + C, ∫ x^3 dx = x^4/4 + C. for a ≠ -1. ∫ x dx = x^2/ 2 + C, ∫ 1 dx = ∫ x^0 dx = x + C. ∫ dx / x = ln|x| + C. ∫ 5 / x dx = 5, ∫ 1 / x dx = 5 * ln|x| + C. ∫ e^x dx = e^x+C, ∫ a^x dx = a^x / ln a + C, ∫ 2^x dx = 2^x / ln2 + C, ∫ 5^x = 5^x / ln5 +C. ∫ f'(x) dx / f(x)} = ln |f(x)| + C, ∫ 2x / (x^2 + 1) dx = ln|x^2 + 1| + C. Integrals of trigonometric functions. ∫ sin x dx = -cos x + C, ∫ tan x dx = -ln|cos x| + C, ∫ dx / cos^2 x = tan x + C. ∫ cos x dx = sin x + C. ∫ cot x dx = ln |sin x| + C, ∫ dx / sin^2 x = -cot x + C.