How to Calculate Integral of Root Root X N 1

What is the Integral of Root Root x n 1?

The integral of root root x n 1 refers to the antiderivative of the function √(√(x^n + 1)). This involves integrating a nested square root function, which requires careful application of integration techniques such as substitution and simplification.

Understanding this integral is useful in various mathematical and scientific applications, including physics, engineering, and economics, where nested root functions appear in models and equations.

Formula for the Integral

The integral of √(√(x^n + 1)) with respect to x can be expressed using the following formula:

This formula is derived using the method of substitution, which simplifies the nested root function into a more manageable form for integration.

How to Calculate the Integral

To calculate the integral of √(√(x^n + 1)), follow these steps:

1. Identify the inner function: √(x^n + 1).
2. Let u = √(x^n + 1). Then, du/dx = (n/2) * x^(n-1).
3. Express dx in terms of du: dx = (2/n) * x^(1-n) du.
4. Substitute u and dx into the integral: ∫u * (2/n) * x^(1-n) du.
5. Simplify the expression and integrate with respect to u.
6. Substitute back to x and add the constant of integration C.

This method ensures that the integral is calculated accurately and efficiently.

Worked Example

Let's calculate the integral of √(√(x^2 + 1)) from 0 to 1.

1. Apply the formula: ∫√(√(x^2 + 1)) dx = (2/5) * (x^(5/4) - 1)^(5/2) + C.
2. Evaluate at the upper limit (x = 1): (2/5) * (1^(5/4) - 1)^(5/2) = (2/5) * (1 - 1)^(5/2) = 0.
3. Evaluate at the lower limit (x = 0): (2/5) * (0^(5/4) - 1)^(5/2) = (2/5) * (-1)^(5/2).
4. Subtract the lower limit from the upper limit: 0 - (2/5) * (-1)^(5/2).
5. The result is (2/5) * (1)^(5/2) = 2/5.

This example demonstrates how to apply the integral formula to a specific case.