Integrating 1/sqrt(x^2-1) Using Trigonometric Substitution

Integration of functions in calculus often involves techniques like substitution, especially when dealing with complex expressions. One such expression is the integral of 1/sqrt(x^2-1). This article will explore how to integrate the function using trigonometric substitution, specifically by letting x secθ. This method simplifies the integration process and provides a clear understanding of the underlying trigonometric relationships.

Introduction to the Integral

The problem at hand is to evaluate the integral I ∫1/sqrt(x^2 - 1)dx. This involves a function that cannot be integrated directly using elementary methods. However, by employing the trigonometric substitution technique, we can simplify the integral significantly.

Trigonometric Substitution: x secθ

First, let's set x secθ. Consequently, the differential dx can be expressed as:

dx secθ tanθ dθ

The integral now takes the form:

I #x222B; 1 #x221E; 1 x 2 - 1 #x2146; x

Substituting x secθ and dx secθ tanθ dθ, the integral becomes:

I #x222B; 1 #x221E; sec θ tan θ dθ tan θ

This simplifies to:

I #x222B; 1 #x221E; sec θ dθ

The integral of secθ is well-known and can be evaluated as:

I ln | sec θ tan θ | C

Since x secθ, we substitute back to get the final answer in terms of x:

I ln | x x 2 - 1 | C

Conclusion

By using the trigonometric substitution technique, the integral of 1/sqrt(x^2-1) has been successfully evaluated. This approach not only simplifies the integrand but also provides a clear understanding of the underlying mathematical relationships.

Related Keywords

integration trigonometric substitution secant function