Calculus 2 — 10.5: Stacked Substitutions for Quadratic Radicals
Evaluate an integral with an ugly quadratic radical by chaining three substitutions in sequence: completing the square, trig substitution, and a tangent substitution. This video shows how each layer converts the problem into a friendlier standard form, then carefully unwinds each substitution back to the original variable.
Key concepts covered:
Completing the square on a general quadratic with a leading negative coefficient
The sign-flip trap when adding inside parentheses preceded by a minus
Recognizing the a² − u² pattern and choosing u = a sin θ
Building and reading a reference triangle for trig substitution
Reducing the integrand to (1/16) ∫ sec⁴θ dθ
Splitting sec⁴θ as sec²θ · sec²θ and substituting w = tan θ
Unwinding substitutions in reverse order: w → θ → u → x
Writing the final answer in terms of the original radicand 3 − 2x − x²
How this technique fits alongside partial fractions in the integration toolkit
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SOURCE MATERIALS
The source materials for this video are from • Calculus 2 Lecture 7.3: Integrals By Trig...