Evaluating a Definite Integral: A Step-by-Step Guide
Evaluating a Definite Integral: A Step-by-Step Guide
In this article, we will delve into the process of evaluating a specific definite integral, which involves a series of sophisticated integration techniques. This integral, represented by the symbol I, is:
What is the Value of the Definite Integral?
We want to evaluate the definite integral:
(I displaystyleint_{2/sqrt{3}}^2 frac{1}{x^2 - 1^{3/2}}, dx)
The integral is nested within a mathematical context, and the solution involves several key steps, including trigonometric substitution and integration by substitution method. Let’s break down each step to understand the process fully.
Step 1: Trigonometric Substitution
The first step is to make a trigonometric substitution. We set:
(x sec{t})
Considering the limits, we have:
When (x frac{2}{sqrt{3}}), (sec{t} frac{2}{sqrt{3}}), so (t frac{pi}{6}).
When (x 2), (sec{t} 2), so (t frac{pi}{3}).
Substituting (x sec{t}) into the integral, we get:
(I displaystyleint_{pi/6}^{pi/3} frac{1}{tan^3{t}} cdot sec{t} tan{t}, dt displaystyleint_{pi/6}^{pi/3} frac{sec{t}}{tan^2{t}}, dt)
Step 2: Rewriting in Terms of Sine and Cosine
Next, we rewrite the integral in terms of sine and cosine:
(I displaystyleint_{pi/6}^{pi/3} frac{frac{1}{cos{t}}}{frac{sin^2{t}}{cos^2{t}}}, dt displaystyleint_{pi/6}^{pi/3} frac{cos{t}}{sin^2{t}}, dt)
Step 3: Further Substitution
We make another substitution to simplify the integral. Let:
(w sin{t})
This implies that:
(dw cos{t}, dt)
And the limits of integration in terms of (w) are:
When (t frac{pi}{6}), (w frac{1}{2}).When (t frac{pi}{3}), (w frac{sqrt{3}}{2}).
Substituting (w sin{t}) and (dw cos{t}, dt) into the integral, we obtain:
(I displaystyleint_{1/2}^{sqrt{3}/2} frac{1}{w^2}, dw)
Step 4: Complete Evaluation
Now, we can integrate and evaluate the limits:
(I -frac{1}{w}Bigg_{1/2}^{sqrt{3}/2})Substituting the limits, we get:
(I -frac{1}{frac{sqrt{3}}{2}} frac{1}{frac{1}{2}}) ( -frac{2}{sqrt{3}} 2 frac{2}{sqrt{3}} - 2)
Conclusion
This demonstrates the power of trigonometric substitutions and integration by substitution in solving complex integrals. The final value of the integral is:
(I frac{2}{sqrt{3}} - 2)
Understanding and applying these techniques can greatly enhance one’s ability to solve advanced mathematical problems.