Solving a Complex Integral Through Substitution and Simplification

Integrals often present challenges, especially when they involve complex expressions under a square root. Let's walk through the process of solving an integral using strategic substitutions and simplification steps. This article covers integrating the expression ∫(x - 1) / (x 1*sqrt(x^3 x^2 x)) dx. We will explore the steps and techniques to make this integral manageable and solvable.

Step 1: Simplifying the Integrand

The first step is to simplify the expression under the square root. Notice that x^3 x^2 x can be factored as x(x^2 x 1).

Thus, we can rewrite the integral as:

I ∫(x - 1) / (x 1*sqrt(x(x^2 x 1))) dx

Step 2: Substitution for Simplification

To simplify the integral, we introduce a substitution:

u sqrt(x(x^2 x 1))

First, we need to compute du/dx.

Calculating:

u^2 x(x^2 x 1)

and differentiating both sides:

2u * du/dx 3x^2 2x 1

Thus,

du/dx (3x^2 2x 1) / (2u)

Reversing the operation and solving for dx:

dx 2u / (3x^2 2x 1) du

Step 3: Substituting Back Into the Integral

We then substitute both the new variable u and the expression for dx into the integral. This step is crucial for transforming the integral into a more manageable form:

I ∫(x - 1) / (x 1*u) * (2u / (3x^2 2x 1) du)

Step 4: Partial Fraction Decomposition (Alternative Method)

Instead of going through complex partial fraction decomposition, we recognize that the integral might be simplified using another substitution. Let's consider:

x - 1 A/sqrt(x^3 x^2 x) B/(x 1)

This step helps in evaluating the integral, but the exact values for A and B might be derived after further manipulation.

Step 5: Final Substitution for Direct Integration

For simplicity, let's use the substitution:

x - 1 t^2

Transforming x and dx:

x t^2 1

dx 2t dt

Substituting back:

I ∫(2t) / (t^2 2t 2) dt

This integral can be solved using standard techniques. By breaking it into simpler parts, we can evaluate:

I 2∫dt / (t 1 1/√2) 2ln|t 1 1/√2| C

Reverting the substitution:

I 2ln|sqrt(x^3 x^2 x) (x 1)/2| C

Conclusion

The integral can be solved using the substitution method or by breaking it into simpler parts. These techniques help in transforming a complex integral into a form that can be evaluated using standard integration rules. Whether you choose to use direct substitution or partial fractions, these methods provide a pathway to finding the solution. By leveraging computational tools, you can also evaluate the integral numerically for specific values of x.