Integral of ∫dx/sqrt(a^2 + x^2)

The integral ∫dx/√(a² + x²) may be approached in 2 ways:

1. Trigonometric substitution, or
2. Hyperbolic substitution

The goal of both methods is to simplify the denominator expression: √(a² + x²).

With trigonometric substitution, we can let x = atanθ, and thus dx = asec²θdθ, and √(a² + x²) simplifies to √[a²(1 + tan²θ)] = asecθ. And the integral becomes ∫secθdθ.

With hyperbolic substitution, we can let x = asinhu, and thus dx = acoshudu, and √(a² + x²) simplifies to √[a²(1 + sinh²u)] = acoshu, so the integral becomes ∫du.

Please give me a "thumbs up" if you have found this video helpful.

Please ask me a maths question by commenting below and I will try to help you in future videos.