Unlock the Mystery: Antiderivative of √x Explained!
Understanding calculus, particularly the concept of integration, often begins with grasping fundamental antiderivatives. Khan Academy provides excellent resources for those seeking clarity on integral calculus. The question of what is the antiderivative of a square root, frequently encountered in problems concerning area under a curve, is a core component of this understanding. Specifically, what is the antiderivative of a square root can be unraveled by understanding the power rule in integration.
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Unlocking the Mystery: Antiderivative of √x Explained!
This explanation delves into finding the antiderivative of the square root function, specifically addressing "what is the antiderivative of a square root". We'll break down the process step-by-step, making it easy to understand.
Understanding Antiderivatives: The Basics
Before tackling the square root, let's ensure a solid foundation in antiderivatives, also known as indefinite integrals.
What is an Antiderivative?
An antiderivative is the reverse operation of a derivative. Think of it as asking: "What function, when differentiated, gives us the function we're starting with?" Finding the antiderivative is called integration. Crucially, antiderivatives aren't unique; they can differ by a constant. This "constant of integration" is always represented as "+ C".
Notation
The integral symbol ∫ represents integration. So, the question "what is the antiderivative of a square root" is mathematically written as:
∫ √x dx
This reads as "the integral of the square root of x with respect to x."
Tackling the Square Root: √x
Now, let's focus on the core of the topic: finding the antiderivative of the square root of x.
Rewriting the Square Root
The first essential step is to rewrite the square root using fractional exponents. Remember that √x is the same as x raised to the power of 1/2:
√x = x1/2
This transformation is crucial because it allows us to apply the power rule of integration.
The Power Rule of Integration
The power rule is a fundamental tool for finding antiderivatives of power functions. It states:
∫ xn dx = (xn+1) / (n+1) + C, where n ≠ -1
Notice that we're adding 1 to the exponent and then dividing by the new exponent. The "+ C" is vital – don't forget it!
Applying the Power Rule to √x
Now, we can apply the power rule to x1/2. Here, n = 1/2.
- Add 1 to the exponent: 1/2 + 1 = 3/2
- Divide by the new exponent: x3/2 / (3/2)
- Simplify the division: Dividing by a fraction is the same as multiplying by its reciprocal. So, x3/2 / (3/2) becomes (2/3)x3/2
- Add the constant of integration: (2/3)x3/2 + C
Therefore, the antiderivative of √x is (2/3)x3/2 + C.
Verification
To ensure the result is correct, we can differentiate the antiderivative and see if we get back to the original function, √x.
Differentiating (2/3)x3/2 + C
Using the power rule of differentiation:
d/dx [(2/3)x3/2 + C] = (2/3) (3/2) x(3/2 - 1) + 0
Simplifying:
= x1/2
= √x
Since differentiating the antiderivative gives us back the original function, √x, the calculated antiderivative is correct.
Summary Table
| Function | Antiderivative |
|---|---|
| √x | (2/3)x3/2 + C |
| x1/2 | (2/3)x3/2 + C |
Video: Unlock the Mystery: Antiderivative of √x Explained!
FAQs: Decoding the Antiderivative of √x
Got questions about finding the antiderivative of the square root of x? Here are some common queries answered:
What's the general formula for finding the antiderivative of √x?
The antiderivative of √x, also written as x^(1/2), is (2/3)x^(3/2) + C, where C is the constant of integration. Remember to always include "+ C" when finding indefinite integrals. This stems from the power rule for integration.
Why do we add "+ C" when finding the antiderivative of √x?
The "+ C" represents the constant of integration. When you differentiate (2/3)x^(3/2), you get √x. However, differentiating (2/3)x^(3/2) + 5 also gives you √x. Therefore, any constant could be present, and "+ C" accounts for all possibilities.
How does this apply to other square root functions?
This principle can be extended to other square root functions, or any function in the form x^n. Just use the power rule for integration: the integral of x^n is (x^(n+1))/(n+1) + C.
Is there another way to express what is the antiderivative of a square root?
Yes, you can also express the antiderivative of √x as (2/3) (√x)^3 + C. Both (2/3)x^(3/2) + C and (2/3) (√x)^3 + C are mathematically equivalent ways of representing what is the antiderivative of a square root. They both represent the same set of functions.