Introduction

Integration is a fundamental concept in calculus that involves finding the area under a curve. This process is essential in various fields such as physics, engineering, and economics. In this article, we will explore the integration of the function (frac{1}{sqrt{x^{10}-x^2}}). We will provide a detailed step-by-step solution to help you understand the process and the underlying techniques involved.

Step-by-Step Solution

I integration

The initial integral is given as:

(I intfrac{dx}{sqrt{x^{10} - x^2}})

This can be simplified as:

(I intfrac{dx}{xsqrt{x^8 - 1}})

We use the substitution:

(x^4 t Longrightarrow 4x^3 dx dt Longrightarrow dx frac{dt}{4x^3})

Substituting this in the integral:

(I int frac{frac{dt}{4x^3}}{x sqrt{x^8 - 1}} frac{1}{4}intfrac{dt}{x^4 sqrt{t^2 - 1}})

We further simplify using the identity (x^4 t):

(I frac{1}{4}intfrac{dt}{tsqrt{t^2 - 1}} frac{1}{4}sec^{-1} t C frac{1}{4}sec^{-1} (x^4) C)

Alternative Solutions

Method 1

The integral can also be approached as:

(I int frac{dx}{sqrt{x^{10} - x^2}})

By substituting:

(x^{-4} t Longrightarrow x^{-5} dx dt)

The integral simplifies to:

(I int frac{dt}{sqrt{1 - t^2}} cos^{-1} t C cos^{-1} (x^{-4}) C)

Method 2

Another alternative approach is:

(I int frac{dx}{sqrt{x^{10} - x^2}} int frac{x^{-5}}{sqrt{1 - x^{-8}}} dx)

By using the same substitution:

(x^{-4} t Longrightarrow x^{-5} dx dt)

The integral simplifies to:

(I int frac{dt}{sqrt{1 - t^2}} cos^{-1} t C cos^{-1} (x^{-4}) C)

Mathematical Notations and Techniques

The key techniques used in these solutions include:

Substitution method: Using a substitution such as (x^4 t) or (x^{-4} t) to simplify the integral. Trigonometric identity: Utilizing the identity (frac{1}{sqrt{1 - t^2}} sec^{-1} t). Integration of standard forms: Recognizing standard integrals such as (int frac{dt}{sqrt{1 - t^2}} cos^{-1} t C).

Conclusion

In conclusion, the integration of the function (frac{1}{sqrt{x^{10} - x^2}}) can be effectively managed using various substitution methods and trigonometric identities. Understanding these techniques is crucial for solving more complex integrals and applying calculus in real-world problems.

Keywords: integration, calculus, mathematical integration