Antiderivative Of CSC Antiderivative Of CSC Before defining the ant derivative of the function first define the term derivative of the function. We know if we draw a graph of a given function and also draw a straight line and that line touches the graph at a point on the graph then this point is define as the derivative of the given function, that means derivative is the process that differentiae the given function at the particular point. These functions are differentiated by using the derivative rules of trigonometry. After defining the differentiation we define the term anti derivative of the function ‘f’ that define as a function ‘F’ which has the derivative ‘f’ so in simple words we can say that the anti derivative of function ‘f’ is the function ‘F’ whose derivative is function ‘f’. Function of differentiation and anti differentiation is opposite to each other. Expression of anti derivative is denoted as F' = f and when we talk about the Ant derivative of trigonometric function as example antiderivative of csc that is also written as cosec or 1 / sin, and it is defined as follows: Know More About Calculus Limits Worksheets
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∫ (1 / sin c) dc = ∫ csc c dc , Multiply and divide by (csc ac– cot c) that gives ∫ (csc c (csc c – cot c) / (csc c - cot c), = ∫ (csc2 c – csc c cot c) / (- cot c + csc c) dc, As we know the function ∫ cscn pc dc = - cscn - 1 pc cos pc / p (n – 1) + (n – 2) / (n – 1) ∫ cscn - 2 pc dc and one thing should be noted that ‘n’ is not equals to -1. = In the numerator we have the derivative function dc so; ∫ d (-cot c + csc c) / (-cot c + csc c), = ln | csc c – cot c | + v. (Here ‘v’ is integral constant and define as the non zero constant). Integral of function ‘csc c’ then, csc c = sec (p / 2) – c, we obtain as ∫ csc c dc = ∫ sec (p / 2) – c, If we take a variable as x = (p / 2) – a so dx = - dc , Read More About Calculus Limit Worksheet
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Now equation is define as - ∫ sec x dx = - log (sec x + tan x) + v, Rewrite the value of ‘x’ is x = (p / 2) – c equation is, - ∫ sec x dx = -log (sec (p / 2) - c + tan (p / 2) – c + v, Formula for integration is written by using the function of trigonometry as ∫ csc c dc = - log (csc c + cot c) + v, equation in form of limits then it is written as: p∫q f (x) = F (y) – F (z). So It can be easily understood that the term anti derivative is the opposite function of derivative that is similar to the integration. If we define the anti derivative of ∫ csc x dx, it is also written in form of integration as ∫ 1 / sin x dx because 1 / sin x = csc x
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