Find the most general antiderivative

Find the most general antiderivative : Square root of 3 Please with clear hand writing, will down vote if I cant understand the numbers and explanation.

The Correct Answer and Explanation is:

Let’s find the most general antiderivative of the constant function: 3\sqrt{3}3​

Step 1: Understand the Function

We’re given a constant function: f(x)=3f(x) = \sqrt{3}f(x)=3​

This function returns the same value (√3) for every value of x. The goal is to find its antiderivative or indefinite integral.

Step 2: Use the Rule for Integrating a Constant

The integral of a constant aaa with respect to xxx is: ∫a dx=ax+C\int a \, dx = ax + C∫adx=ax+C

where CCC is the constant of integration.

Step 3: Apply the Rule

Apply the rule to our specific constant 3\sqrt{3}3​: ∫3 dx=3x+C\int \sqrt{3} \, dx = \sqrt{3}x + C∫3​dx=3​x+C

✅ Final Answer:

∫3 dx=3x+C\boxed{\int \sqrt{3} \, dx = \sqrt{3}x + C}∫3​dx=3​x+C​

Explanation

An antiderivative of a function is another function whose derivative gives the original function. In this case, we are asked to find the most general antiderivative of the constant 3\sqrt{3}3​.

The constant 3\sqrt{3}3​, which is approximately 1.732, is a real number and does not depend on the variable xxx. So when we integrate it with respect to xxx, we are essentially summing the constant value over a range of xxx values. That means we are looking for a function whose derivative is a constant.

The derivative of 3x\sqrt{3}x3​x is 3\sqrt{3}3​. Therefore, 3x\sqrt{3}x3​x is an antiderivative of 3\sqrt{3}3​. But to express the most general antiderivative, we must also include an arbitrary constant CCC, because the derivative of any constant is zero. Adding any constant to 3x\sqrt{3}x3​x will still result in a derivative of 3\sqrt{3}3​.

That is why the answer is: 3x+C\sqrt{3}x + C3​x+C

This formula captures all possible antiderivatives of the function f(x)=3f(x) = \sqrt{3}f(x)=3​. It includes every vertical shift of the line 3x\sqrt{3}x3​x, each of which has the same constant slope and thus the same derivative.

Including the constant CCC is important in indefinite integrals because differentiation loses constant information. Integration restores the family of functions that could have led to the same derivative.

So, the most general antiderivative of 3\sqrt{3}3​ is: 3x+C\boxed{\sqrt{3}x + C}3​x+C​