The Integral of x/√x: A Complete Guide to Solving This Common Calculus Problem
The integral of x divided by the square root of x is one of those problems that looks intimidating at first glance but becomes remarkably straightforward once you apply the right simplification techniques. Consider this: whether you are a student encountering calculus for the first time or a seasoned learner looking to refresh your skills, understanding how to evaluate ∫ x/√x dx is an essential building block in mastering integral calculus. In this article, we will walk through every step of solving this integral, explore the underlying mathematical principles, and provide clear examples to ensure you have a thorough understanding of the process.
Understanding the Problem
Before diving into the solution, let us first clarify what we are being asked to compute. And the expression ∫ x/√x dx represents the indefinite integral of the function f(x) = x/√x with respect to x. The goal is to find a function F(x) such that its derivative F'(x) equals x/√x.
At first glance, this looks like it might require advanced integration techniques such as integration by parts or substitution. Even so, the beauty of this problem lies in its simplicity once you recognize that the integrand can be greatly simplified using basic exponent rules.
Simplifying the Integrand
The key insight here is to rewrite the expression x/√x using exponent notation. Recall that the square root of x can be written as:
√x = x^(1/2)
That's why, the fraction x/√x can be rewritten as:
x / x^(1/2) = x^(1) · x^(-1/2) = x^(1 - 1/2) = x^(1/2)
This simplification is based on the fundamental exponent rule which states that when dividing powers with the same base, you subtract the exponents:
x^a / x^b = x^(a-b)
So the original integral simplifies dramatically:
∫ x/√x dx = ∫ x^(1/2) dx = ∫ √x dx
This is now a simple power function that we can integrate using the power rule for integration.
Applying the Power Rule for Integration
The power rule for integration is one of the most fundamental tools in calculus. It states that for any real number n ≠ -1:
∫ x^n dx = (x^(n+1)) / (n+1) + C
where C is the constant of integration.
In our case, n = 1/2. Applying the power rule:
∫ x^(1/2) dx = (x^(1/2 + 1)) / (1/2 + 1) + C
Now let us compute the exponent and the denominator separately:
- The new exponent: 1/2 + 1 = 1/2 + 2/2 = 3/2
- The new denominator: 1/2 + 1 = 3/2
Substituting these values back in:
∫ x^(1/2) dx = (x^(3/2)) / (3/2) + C
Dividing by a fraction is the same as multiplying by its reciprocal:
(x^(3/2)) / (3/2) = x^(3/2) · (2/3) = (2/3) · x^(3/2)
Because of this, the final answer is:
∫ x/√x dx = (2/3) · x^(3/2) + C
Verifying the Result
One of the best habits in calculus is to always verify your answer by differentiating the result. If we differentiate (2/3) · x^(3/2) + C, we should get back our original integrand x/√x Which is the point..
Using the power rule for differentiation:
d/dx [(2/3) · x^(3/2)] = (2/3) · (3/2) · x^(3/2 - 1) = (2/3) · (3/2) · x^(1/2) = 1 · x^(1/2) = √x
And since √x = x/√x · (√x/√x) ... well, more directly:
√x = x^(1/2) = x / x^(1/2) = x / √x
This confirms that our integration was correct. The derivative of our answer returns us to the original function, which is the hallmark of a correct integration And that's really what it comes down to..
Alternative Approach: Using Substitution
While the simplification method described above is the most efficient approach, it is also worth exploring how substitution would work for this integral, as it provides valuable practice for more complex problems That's the whole idea..
Let us set u = √x, which means:
u = x^(1/2)
Taking the derivative of both sides with respect to x:
du/dx = (1/2) · x^(-1/2) = 1/(2√x)
Rearranging to solve for dx:
dx = 2√x · du = 2u · du
Now we also need to express x in terms of u:
x = u^2
Substituting everything into the original integral:
∫ x/√x dx = ∫ (u^2 / u) · (2u) du = ∫ u · 2u du = ∫ 2u^2 du
Now applying the power rule:
∫ 2u^2 du = 2 · (u^3/3) + C = (2/3)u^3 + C
Substituting back u = √x:
(2/3)(√x)^3 + C = (2/3) · x^(3/2) + C
This confirms our earlier result. Both methods yield the same answer, which is reassuring and demonstrates the consistency of calculus Worth knowing..
Geometric Interpretation
To deepen your understanding, it is helpful to think
