A trigonometric substitution converts a square-root identity into a right triangle.
WHAT HAPPENS IF…Why is the angle interval important?
Interactive exploration
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Connect the picture to the ideax=asinθ⟹a2−x2=acosθ
For a>0 and −π/2<θ<π/2, the identity 1−sin²θ=cos²θ makes the root a cosθ. Also dx=a cosθ dθ, so ∫dx/√(a²−x²)=∫dθ=arcsin(x/a)+C. The domain restriction justifies the sign choice.
A detail that matters
Taking a square root requires the nonnegative branch.