Antiderivative Of Sin2x Antiderivative Of Sin2x The definition of Antiderivative is, ∫ g (x) dx = f(x )+ c, where d f(x) /dx = g(x). The following are the integrals of the trigonometric functions. Following are the integrals or antiderivatives of the sin, cos, tan, cot, cosec etc. functions. For general if the sin x is a trigonometric function then cos x is the derivative of that function. Antiderivatives of some of the sin function are as follows- ∫ sin ax = - (1/a) cos ax + c ∫ sin n ax dx = - sin (n-1) . ax . cos ax / na + (n-1 )/ n . ∫ sin n-2 ax dx cos ax / na + (n-1 )/ n . ∫ sin n-2 ax dx ( for n > 2 ) Antiderivative of the two cos functions are as following- ∫ cos ax dx = (1/a) sin ax +c ∫ cosn ax dx = cos n-1 ax . sin ax / na + ( n-1 / n ) . ∫ cos n-2 ax dx (for n > 0) Integral of the tan x is defined as the formula below ∫ tan ax dx = -( 1 / a ) log ( cos ax ) + c Antiderivative of the \secant function is defined as follows- ∫ sec ax dx = ( 1 / a ) log (sec ax + tan ax ) + c Integral of the cosec x is as follows- ∫ cosec ax dx = -( 1 / a ) log (cosec ax + cot ax ) + c Integral of the cot x is as follows- ∫ cot ax dx = ( 1 / a ) sin ax +c Here c is the integral constant in all the trigonometric functions defined above and a is a constant and n is the positive integral.
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In calculus, an antiderivative, primitive integral or indefinite integral[1] of a function f is a function F whose derivative is equal to f, i.e., F ′ = f. The process of solving for antiderivatives is called antidifferentiation (or indefinite integration) and its opposite operation is called differentiation, which is the process of finding a derivative. Antiderivatives are related to definite integrals through the fundamental theorem of calculus: the definite integral of a function over an interval is equal to the difference between the values of an antiderivative evaluated at the endpoints of the interval. The function F(x) = x3/3 is an antiderivative of f(x) = x2. As the derivative of a constant is zero, x2 will have an infinite number of antiderivatives; such as (x3/3) + 0, (x3/3) + 7, (x3/3) − 42, (x3/3) + 293 etc. Thus, all the antiderivatives of x2 can be obtained by changing the value of C in F(x) = (x3/3) + C; where C is an arbitrary constant known as the constant of integration. Essentially, the graphs of antiderivatives of a given function are vertical translations of each other; each graph's location depending upon the value of C. In physics, the integration of acceleration yields velocity plus a constant. The constant is the initial velocity term that would be lost upon taking the derivative of velocity because the derivative of a constant term is zero. This same pattern applies to further integrations and derivatives of motion (position, velocity, acceleration, and so on).
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If F is an antiderivative of f, and the function f is defined on some interval, then every other antiderivative G of f differs from F by a constant: there exists a number C such that G(x) = F(x) + C for all x. C is called the arbitrary constant of integration. If the domain of F is a disjoint union of two or more intervals, then a different constant of integration may be chosen for each of the intervals. For instance is the most general antiderivative of on its natural domain Every continuous function f has an antiderivative, and one antiderivative F is given by the definite integral of f with variable upper boundary: Varying the lower boundary produces other antiderivatives (but not necessarily all possible antiderivatives). This is another formulation of the fundamental theorem of calculus. There are many functions whose antiderivatives, even though they exist, cannot be expressed in terms of elementary functions (like polynomials, exponential functions, logarithms, trigonometric functions, inverse trigonometric functions and their combinations). Examples of these are
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