Evaluating Integrals: Techniques and Applications

Evaluating integrals is a critical skill in calculus, often used across various fields such as physics, engineering, and economics. In this article, we will explore a specific example of evaluating an integral using substitution methods. Specifically, we will look at the integral:

I displaystyle int frac{dx}{xsqrt{x^2 - x - 1}}

Introduction to the Integral

The integral in question appears complex, involving a square root in the denominator. However, by applying appropriate substitutions, we can simplify and evaluate it. Let's walk through the process step by step.

Step 1: Substitution to Simplify the Expression

First, we introduce a substitution to simplify the expression under the square root:

u x^2 - x - 1

Then, we compute the derivative:

du (2x - 1)dx

From this, we express dx in terms of du:

dx frac{du}{2x - 1}

Next, we express x in terms of u using the quadratic formula:

x frac{-1 pm sqrt{1 - 4(1 - u)}}{2} frac{-1 pm sqrt{4u - 3}}{2}

Step 2: Express x - 1 in Terms of u

We need to express x - 1 in terms of u:

x - 1 frac{1 - sqrt{4u - 3}}{2}

Step 3: Substituting Back into the Integral

Now we substitute dx and x - 1 into the original integral:

I int frac{1}{left(frac{1 - sqrt{4u - 3}}{2}right)sqrt{u}} cdot frac{du}{2x - 1}

We need to express 2x - 1 in terms of u:

2x - 1 -1 - sqrt{4u - 3}

Thus, the integral becomes:

I int frac{2}{(1 - sqrt{4u - 3})sqrt{u}} cdot frac{du}{sqrt{4u - 3}}

Simplifying, we get:

I int frac{2}{(1 - sqrt{4u - 3})(4u - 3)} du

Step 4: Alternative Substitution Using Trigonometric Identities

Consider another substitution, x tan t - 1:

dx sec^2 t dt

With this substitution, we have:

sqrt{x^2 - x - 1} sqrt{tan^2 t - 2tan t 1 - tan t - 1} sqrt{tan^2 t - tan t}

The integral now becomes:

I 2 int frac{sec^2 t}{tan t sqrt{tan^2 t - tan t}} dt

Using trigonometric identities, we can simplify this further:

I 2 int frac{dt}{sqrt{3}sin t cos t}

This can be rewritten as:

I int frac{dt}{cos t - frac{pi}{3}} ln left(sec t - frac{pi}{3}right)tan t - frac{pi}{3}

Final Evaluation

After performing the necessary algebra and integration techniques, we arrive at:

I 2 arcsin left(frac{x}{sqrt{x^2 - x - 1}}right) C

Where C is the constant of integration.

Conclusion

Evaluating integrals can be a challenging but rewarding process. By understanding and applying different substitution methods, we can simplify and solve seemingly complex integrals. In this article, we explored the integral involving a square root in the denominator and used both algebraic and trigonometric substitutions to evaluate it.