Calculating Integrals with Substitution and Trigonometric Identities

When it comes to solving integrals, a combination of substitution and trigonometric identities can be powerful tools. In this article, we will explore a particular example. We will walk through the process of solving an integral step-by-step, detailing how to apply these techniques to reach a solution.

Introduction to the Integral Problem

We aim to evaluate the integral:

I ∫01/2 (1/√3) / (x^2√(1-x^3)) dx

Step-by-Step Solution

Simplifying the Integral

First, we need to simplify the given integral by recognizing that the expression under the fourth root can be rewritten as:

∫01/2 (1/√3) / (x^2√(1-x^3)) dx ∫01/2 (1/√3) / (√x(1-x^3)) dx

Using Substitution

To solve the integral, we can use the substitution method. Let's set:

x (1-t^2)/2, so dx -t dt

When x 0, t sqrt(3), and when x 1/2, t 1.

Transforming the Integral

By substituting the expression for x and dx, the integral becomes:

I ∫1sqrt(3) (1/√3) dt (2/√3) * (sqrt(3) - 1) 2

Alternative Solution Method

Alternatively, let's consider another approach involving trigonometric substitution:

Using Trigonometric Substitution

Let us make the substitution:

x sin θ, so dx cos θ dθ

When x 0, θ 0, and when x 1/2, θ π/6.

Transforming the Integral

Thus, the integral can be rewritten as:

I (1/√3) ∫0π/6 (1 / (1 - sin θ) cos θ) dθ

Further Simplification

This integral can be split into two parts:

I (1/√3) [ ∫0π/6 (1 / cos2θ) dθ - ∫0π/6 (sin θ / cos2θ) dθ ]

The first integral is the secant squared function, while the second involves the tangent function:

I (1/√3) [ tan θ |0π/6 - (1/ cos θ) |0π/6 ]

By evaluating at the limits:

I (1/√3) [ (√3/3) - 1 ]

2

Conclusion

Both methods lead to the same result, demonstrate the effectiveness of substitution and trigonometric identities in solving complex integrals. This process enriches our understanding of integral calculus and provides a versatile approach to problem-solving.