Calculus
Calculus is the mathematical study of change,[1] in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations. It has two major branches, differential calculus (concerning rates of change and slopes of curves), and integralcalculus (concerning accumulation of quantities and the areas under curves); these two branches are related to each other by the fundamental theorem of calculus. Both branches make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-definedlimit. Calculus has widespread uses in science, economics, and engineering and can solve many problems that algebra alone cannot. Calculus is concerned with comparing quantities which vary in a non-linear way. It is used extensively in science and engineering since many of the things we are studying (like velocity, acceleration, current in a circuit) do not behave in a simple, linear fashion. If quantities are continually changing, we need calculus to study what is going on. BRANCHES OF CALCULUS Calculus is concerned with comparing quantities which vary in a non-linear way. It is used extensively in science and engineering since many of the things we are studying (like velocity, acceleration, current in a circuit) do not behave in a simple, linear fashion. If quantities are continually changing, we need calculus to study what is going on.
Integral Calculus
Indefinite Integrals
Definition: A function F(x) is the antiderivative of a function ƒ(x) if for all x in the domain of ƒ,
F'(x) = ƒ(x) ƒ(x) dx = F(x) + C, where C is a constant.
Basic Integration Formulas
General and Logarithmic Integrals 1. kƒ(x) dx = k ƒ(x) dx | 2. [ƒ(x)g(x)] dx = ƒ(x) dxg(x) dx | 3. k dx = kx + C | 4. xn dx = + C, n-1 | 5. ex dx = ex + C | 6. ax dx = + C, a 0, a1 | 7. = ln |x| + C | |
Trigonometric Integrals 1. sin x dx = -cos x + C | 2. cos x dx = sin x + C | 3.
Integral Calculus
Indefinite Integrals
Definition: A function F(x) is the antiderivative of a function ƒ(x) if for all x in the domain of ƒ,
F'(x) = ƒ(x) ƒ(x) dx = F(x) + C, where C is a constant.
Basic Integration Formulas
General and Logarithmic Integrals 1. kƒ(x) dx = k ƒ(x) dx | 2. [ƒ(x)g(x)] dx = ƒ(x) dxg(x) dx | 3. k dx = kx + C | 4. xn dx = + C, n-1 | 5. ex dx = ex + C | 6. ax dx = + C, a 0, a1 | 7. = ln |x| + C | |
Trigonometric Integrals 1. sin x dx = -cos x + C | 2. cos x dx = sin x + C | 3.