What is integral of root x? ∫√x.dx=? Root x integrate
Understanding the Integral of √x
Introduction
In calculus, the concept of integration is pivotal for understanding areas under curves, solving differential equations, and much more. One common integral that students encounter is ∫√x dx. In this article, we will delve into the process of integrating the square root of x, step-by-step.
What does ∫√x dx mean?
The expression ∫√x dx represents the integral of the function √x with respect to x. The integral aims to find a function whose derivative returns the original function, or geometrically, the area under the curve of √x from a specific point.
Rewriting the Integral
To facilitate the integration process, we can rewrite the square root in terms of exponents:
$√x = x^{1/2}$
This means our integral can be rewritten as:
$∫√x dx = ∫x^{1/2} dx$
Applying the Power Rule of Integration
The power rule of integration states that:
$∫x^n dx = frac{x^{n+1}}{n+1} + C quad (n ≠ -1)$
In our case, n = 1/2. Thus, applying the power rule, we have:
$∫x^{1/2} dx = frac{x^{1/2 + 1}}{1/2 + 1} + C$
Calculating the exponent, we get:
$1/2 + 1 = 3/2$
So,
$∫x^{1/2} dx = frac{x^{3/2}}{3/2} + C$
Now, simplifying (frac{1}{3/2}) yields (frac{2}{3}):
$∫x^{1/2} dx = frac{2}{3} x^{3/2} + C$
Final Result
Thus, the integral of the square root of x is:
$∫√x dx = frac{2}{3} x^{3/2} + C$
where C is the constant of integration, representing an arbitrary constant that could be added to the function.
Conclusion
The integral of √x gives us a clear example of how to use the power rule for integration effectively. Understanding such integrals is crucial for advancing in calculus, as they lay the groundwork for more complex integration techniques and applications in various fields, such as physics, engineering, and economics. The process not only provides valuable skills in mathematical reasoning but also enhances problem-solving capabilities in real-world situations.
