Solving Complex Integral Using Substitution and Partial Fractions

Welcome to this detailed guide on solving a complex integral. This article will explain the step-by-step process of solving the integral ∫(dx / (x - 1)^(3/4) * x^2^(1/4)). We will use various methods including substitution, partial fractions, and integration by parts to derive the final solution.

Problem Statement

Given the integral:

I ∫ (dx / ((x - 1)^(3/4) * x^2^(1/4)))

This integral can be rewritten in a more readable form as:

I ∫ (dx / (sqrt[4]{(x - 1)^3} * x^2^(1/4)))

Step 1: Initial Transformation

Let's first transform the given integral. We start by letting:

y (x - 1)^(1/4)

This implies:

dy (1/4) * (x - 1)^(-3/4) * dx

Substituting these into the integral, we get:

I 4 ∫ (dy / (x^2^(1/4) * y^3)}

Step 2: Further Simplification

We can simplify the expression inside the integral:

I 4 ∫ (dy / (x^2^(1/4) * y^3)}

To proceed, we let:

y (3)^(1/4) * sqrt[tan(θ)]}

This implies:

dy (1/2) * (3)^(1/4) * sec^2(θ) * dθ / sqrt[tan(θ)]}

Step 3: Integration by Parts

Using the new substitution, we can further simplify and integrate:

J (1/6) ∫ (cos(θ) dθ / sqrt[sin(θ)])

Integrating this, we get:

J (1/3) * sqrt[sin(θ)] C}

Substituting back, we obtain:

I (1/3) * sqrt[sqrt[(x - 1) / x^2]] C}

Thus, the solution to the integral is:

I (4/3) * sqrt[4]{(x - 1) / x^2} C}

Conclusion

In conclusion, we have solved the complex integral using substitution, partial fractions, and integration by parts. The final solution is: