Solving Complex Integrals with Substitution and Pattern Recognition

Integral calculus is a significant component of mathematical analysis. One of the fundamental techniques to solve integrals is through the substitution method, which simplifies the original function into a more manageable form. This article explores the application of these techniques, exemplified by two complex integrals. By utilizing substitution and recognizing patterns, we can derive solutions that would initially appear daunting.

The Integral int frac{5x^4 - 4x^5}{x^5 - x 1^2} dx

Let's evaluate the following integral:

(int frac{5x^4 - 4x^5}{x^5 - x 1^2} dx)

To begin with, we employ the substitution method. Let:

(u x^5 - x 1)

Next, we compute the derivative of (u):

(frac{du}{dx} 5x^4 - 1)

From this, we can express (dx) in terms of (du):

(dx frac{du}{5x^4 - 1})

Notice that we can rewrite the integral in terms of (u):

(int frac{5x^4 - 4x^5}{u^2} cdot frac{du}{5x^4 - 1})

However, we need to express (4x^5) in terms of (u) and (x). We can rearrange our substitution:

(x^5 u - x - 1)

Thus, (4x^5 4u - x - 1 4u - 4x - 4)

Substituting this back into our integral, we get a complex expression. Instead of further complicating this, we opt for a simpler approach by recognizing patterns or patterns that can simplify our integral.

We separate the integral into two parts:

(int frac{5x^4}{x^5 - x 1^2} dx int frac{-4x^5}{x^5 - x 1^2} dx)

The first integral can be directly integrated as follows:

(int frac{5x^4}{u^2} cdot frac{du}{5x^4 - 1})

The second integral can be simplified similarly. By performing careful manipulation of these integrals and using partial fractions, we conclude that the integral can be expressed as:

(-frac{1}{u} C -frac{1}{x^5 - x 1} C)

Thus, the final result is:

(int frac{5x^4 - 4x^5}{x^5 - x 1^2} dx -frac{1}{x^5 - x 1} C)

The Integral (int frac{2x^{12} - 5x^9}{x^5 - x^3 1^3} dx)

As a further exercise, consider solving the integral:

(int frac{2x^{12} - 5x^9}{x^5 - x^3 1^3} dx)

Notice that the (x^3 - 1) term in the denominator and the (2x^{12} - 5x^9) term in the numerator have a pattern involving consecutive natural numbers 3 and 6 as the powers of (x).

Let's rewrite the integral:

(int frac{2x^{12} - 5x^9}{x^5 - x^3 1^3} dx int frac{2x^{-3} - 5x^{-6}}{1 - dfrac{1}{x^2} - dfrac{1}{x^3}} dx)

Now, let:

(t 1 - frac{1}{x^2} - frac{1}{x^3}) implies (dt left(-2x^{-3} - 3x^{-4}right)dx) implies (-dt 2x^{-3} 3x^{-4}dx)

The integral simplifies to:

(int frac{-1}{t^2} dt -frac{1}{t} k -frac{1}{1 - frac{1}{x^2} - frac{1}{x^3}} k) where (k in mathbb{R})

As an additional bonus, you can solve this one:

Further Practice

Can you solve the integral:

(int frac{2x^{12} - 5x^9}{x^5 - x^3 1^3} dx)

And don't forget to check out our detailed explanation and solution above to ensure you grasp the methods used.