what is the antiderivative of x 1 What is the antiderivative of tan x dx The antiderivative of tan x dx is tan x ln cos x C What is the antiderivative of 1 x 2 dx 1 x C is the antiderivative of 1 x 2
The antiderivative of frac 1x is the function whose inverse is exactly equal to its own derivative Indeed let y x be the antiderivative of frac 1x Then we have frac dy dx frac 1x Now invert thinking of the Leibniz notation frac dy dx as a rate of change frac dx dy x This means that that frac d dx x x Integration goes the other way the integral or antiderivative of 1 x should be a function whose derivative is 1 x As we just saw this is ln x However if x is negative then ln x is undefined The solution is quite simple the antiderivative of 1 x is ln x Created by Sal Khan Questions
what is the antiderivative of x 1
what is the antiderivative of x 1
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Solved Find The Most General Antiderivative Of The Function Chegg
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6 8 Finding Antiderivatives And Indefinite Integrals Part 5
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dfrac d dx ln x dfrac 1 x nonumber Thus F x ln x is an antiderivative of dfrac 1 x Therefore every antiderivative of dfrac 1 x is of the form ln x C for some constant C and every function of the form ln x C is an antiderivative of dfrac 1 x c We have Thus latex F left x right text ln x latex is an antiderivative of latex frac 1 x latex Therefore every antiderivative of latex frac 1 x latex is of the form latex text ln x C latex for some constant latex C latex and every function of the form latex text ln x C latex is an antiderivative of latex frac 1 x
Thus F x ln x F x ln x is an antiderivative of 1 x 1 x Therefore every antiderivative of 1 x 1 x is of the form ln x C ln x C for some constant C C and every function of the form ln x C ln x C is an antiderivative of 1 x 1 x We have Derivative with respect to x of pi of a constant is just 0 Derivative with respect to x of 1 is just a constant is just 0 So once again this is just going to be equal to 2x In general the derivative with respect to x of x squared plus any constant is going to be equal to 2x
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Using these facts we can find the antiderivative of any polynomial How The fact that the derivative of x k is kx k 1 is equivalent to the statement that the antiderivative of kx k 1 is x k c This means that the antiderivative of x k is frac x k 1 k 1 c What s with this c stuff In order to find an antiderivative for f x frac 1 x we need to remember that dfrac d dx log x frac 1 x text and more generally that dfrac d dx log x frac 1 x text See Example 2 10 4 So begin gather text if f x frac 1 x text then F x log x c end gather
Definition PageIndex 1 Antiderivatives and Indefinite Integrals Let a function f x be given An antiderivative of f x is a function F x such that F x f x The set of all antiderivatives of f x is the indefinite integral of Calculus Find the Antiderivative 1 x 1 x 1 x Write 1 x 1 x as a function f x 1 x f x 1 x The function F x F x can be found by finding the indefinite integral of the derivative f x f x F x f x dx F x f x d x Set up the integral to solve F x
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what is the antiderivative of x 1 - dfrac d dx ln x dfrac 1 x nonumber Thus F x ln x is an antiderivative of dfrac 1 x Therefore every antiderivative of dfrac 1 x is of the form ln x C for some constant C and every function of the form ln x C is an antiderivative of dfrac 1 x c We have