Trigonometric Substitution and Branches — Cheat sheet

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Core rule

Match $a^2-x^2$ , $a^2+x^2$ , or $x^2-a^2$ to sine, tangent, or a branch-aware secant substitution. Replace both the radical and differential.

$\sqrt{u^2}=|u|$ . Complete the square first when needed, state the angle branch, and transform bounds or return to $x$ before evaluation.