what is the antiderivative of the natural logarithm of x The derivative of the logarithm ln x is frac 1 x but what is the antiderivative This turns out to be a little trickier and has to be done using a clever integration by parts
The natural logarithm base e logarithm of a positive real number x represented by lnx or log e x is the exponent to which the base e 2 718 Euler s number is raised to obtain x Mathematically ln x log e x y if Compute answers using Wolfram s breakthrough technology knowledgebase relied on by millions of students professionals For math science nutrition history geography
what is the antiderivative of the natural logarithm of x
what is the antiderivative of the natural logarithm of x
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Suppose the argument of the natural log is not just x but instead is g x a differentiable function Now using the chain rule we get a more general derivative for all The natural logarithm inherits many properties of general logarithms 3 So for all x y 0 the following hold elogx x for any real number X log eX X for any a 1 logax logx log a
The natural logarithm lnx is the logarithm having base e where e 2 718281828 1 This function can be defined lnx int 1 x dt t 2 for x 0 This definition means that e is the unique number with the property that the Derivative of the natural logarithm Let f x ex so that c x and 1 ln Then 1 xx f ex f dx d dx d x 1 1 1 ln c 1 ln Derivative of the logarithm base b After performing a change of base we
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Now the antiderivative rule of power of x is given by x n dx x n 1 n 1 C where n 1 This rule is commonly known as the antiderivative power rule Let us consider some of the examples of this antiderivative rule to understand this The integral also called antiderivative of a function is the reverse process of differentiation
a The derivative of y xn is y0 nx n 1 for n integer b The integral of y x nis Z x dx x n 1 n 1 for n 6 1 c Case n 1 Z dx x is neither rational nor trigonometric function The antiderivative of the natural logarithm ln x is 39 Related formulas such as antiderivatives of logarithms to other bases can be derived from this equation using the
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what is the antiderivative of the natural logarithm of x - Suppose the argument of the natural log is not just x but instead is g x a differentiable function Now using the chain rule we get a more general derivative for all