Evaluating the Integral of 1/(1-x^4) from 0 to Infinity: A Step-by-Step Guide for SEO

When dealing with mathematical integrals, especially those involving infinite limits, it's crucial to approach them methodically and accurately. In this guide, we will break down the process of evaluating the integral of 1/(1-x^4) from 0 to infinity. This integral is a classic example of an improper integral and is of great interest in both mathematics and applications such as physics and engineering.

Step 1: Understanding the Problem

We are tasked with evaluating the following integral:

[ int_0^{infty} frac{1}{1-x^4} , dx ]

Step 2: Expressing the Integrand

To simplify the integrand, we express 1/(1-x^4) in a more manageable form. Notice that 1 can be written as 1/2 times (1 - x^2) (1 - x^2). Let's rewrite the integrand accordingly:

1 1/2 (1 - x^2) (1 - x^2)

Step 3: Applying Partial Fraction Decomposition

We decompose the integrand as follows:

[ frac{1}{1-x^4} frac{1}{2} cdot frac{1-x^2 1-x^2}{1-x^4} frac{1}{2} left( frac{1-x^2}{1-x^4} frac{1-x^2}{1-x^4} right) ]

This can be further simplified to:

[ frac{1}{2} left( frac{1 x^2}{1-x^4} frac{1-x^2}{1-x^4} right) frac{1}{2} left( frac{1 x^2}{1-x^2} frac{1-x^2}{1-x^2} right) ]

Step 4: Simplifying Further

We can now separate the integrand into two components:

[ frac{1}{1-x^4} frac{1 x^2}{2(1-x^2)} frac{1-x^2}{2(1-x^2)} frac{1 x^2}{2(1-x^2)} frac{1-x^2}{2(1-x^2)} ]

This can be further simplified to:

[ frac{1}{2} left( frac{1}{1-x^2} frac{1}{1 x^2} right) ]

Now we have:

[ int_0^{infty} frac{1}{1-x^4} , dx frac{1}{2} int_0^{infty} left( frac{1}{1-x^2} frac{1}{1 x^2} right) , dx ]

Step 5: Evaluating the Integrals

We evaluate the two separate integrals:

[ frac{1}{2} left( int_0^{infty} frac{1}{1-x^2} , dx int_0^{infty} frac{1}{1 x^2} , dx right) ]

The first integral is related to the natural logarithm function:

[ int_0^{infty} frac{1}{1-x^2} , dx left[ frac{1}{2} ln left| frac{1 x}{1-x} right| right]_0^{infty} ]

The second integral is related to the arctangent function:

[ int_0^{infty} frac{1}{1 x^2} , dx left[ tan^{-1}(x) right]_0^{infty} ]

Step 6: Solving the Limits

Evaluating these limits, we get:

[ left[ frac{1}{2} ln left| frac{1 x}{1-x} right| right]_0^{infty} 0 - 0 0 ] [ left[ tan^{-1}(x) right]_0^{infty} frac{pi}{2} - 0 frac{pi}{2} ]

Therefore, combining these results, we have:

[ frac{1}{2} left( 0 frac{pi}{2} right) frac{pi}{4} ]

Conclusion

Thus, the integral of 1/(1-x^4) from 0 to infinity is:

[ int_0^{infty} frac{1}{1-x^4} , dx frac{pi}{4} ]

Understanding this process and the application of mathematical techniques such as partial fraction decomposition and recognizing standard integral forms is crucial for those working with improper integrals and related mathematical problems. This step-by-step guide should help in optimizing your SEO efforts when discussing such integrals and related mathematical concepts.