Give the antiderivatives of 1/x.

Question

Accepted Answer

Welcome back, everyone. Find the family of an inside derivatives capital F of X of the function f of X equals 5 divided by X where X is greater than 0. If F of X represents the family of inside derivatives of the function F of X, specifically lowercase F of X, then we must satisfy the relationship. F of X must be equal to lowercase f x. In other words, if we find a derivative of an inside derivative, then we must return the function f of X. So we can show that F of X is equal to 5 divided by X. This is what the problem suggests, and we can factor out the constant, which is 5, so we get 5 multiplied by 1 divided by X. And then we can focus on the x term. Specifically, we have a fraction 1 divided by X, and we can recall that the derivative of the natural logarithm is equal to 1 divided by X. So what we can do is simply multiply both sides by 5, right? 5 multiplied by the derivative of LN of X is 1 divided by X multiplied by 5. So 5 LN of X is equal to 5 divided by X. This is exactly what we have on the right hand side. And now we're using the properties of derivatives, we can also factor out the constant. It's already factored. So applying the property in reverse, we can show that the derivative of the whole 5 LN of X. Is equal to 5 divided by X. So we have found one of the antiderivatives in order to find the family of anti derivatives. Let's recall that the derivative of any constant is zero. So we're going to add a constant C to our anti derivative 5 LN X to identify the whole family. So that'd be 5 LN of X plus C. The derivative of that is always equal to 5 divided by X, right? X is greater than 0 because it satisfies the domain of the natural log. Therefore, We have identified F of X and we can show that F of X is simply our function 5. LN of X + C. This is our final answer for this problem, and thank you for watching.

Give the antiderivatives of 1/x.

Key Concepts

Antiderivative

Natural Logarithm

Integration Constant

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