Trigonometric Function Substitutions

Calculus 2 · Axiom Academy

LESSON Trigonometric Substitution Turn a stubborn radical into a clean trig function by choosing the right angle — then ride a right triangle back to x . 1. Three Radicals, Three Substitutions Every trig substitution is triggered by the shape of the radical . Each form pairs with the substitution whose Pythagorean identity collapses it to one clean trig factor — watch the square root dissolve as the matching x slides in. 2. The Reference Triangle Makes the Radical a Side The substitution isn't magic — it's a right triangle. Set , label the opposite leg x and the hypotenuse a , and the third side is exactly the radical . That triangle is what lets you translate every trig answer back into x at the end. The Pythagorean theorem fills in the adjacent leg, and reading the triangle gives every other ratio you'll need. Take . The radical is the form with a=3 , so we let , . Watch each line replace the one above as the radical cancels. Substitute, simplify, integrate 4. Finishing: dx and the Trip Back to x Two pieces complete every trig substitution. First, the differential: forces , which is what cancelled the radical above. Second, the answer lives in — so we read the reference triangle backward to return to x . You can now spot which radical calls for sine, tangent, or secant, build the triangle that turns the square root into a side, and ride it back to x . Scroll up to revisit any step.

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