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Calculus 12
4. Integration
4.3 Trigonometric substitution
Trigonometric Substitution
An integration technique for expressions with square roots, replacing the variable with a trig function.
What You'll Learn
Trigonometric substitution integrates expressions with a square root of squares.
Use x = a sine theta for the square root of a squared minus x squared.
Use x = a tangent theta for the square root of a squared plus x squared.
Use x = a secant theta for the square root of x squared minus a squared.
Substitute, simplify with the Pythagorean identity, integrate, then convert back using a triangle.
What You'll Practice
1
Integrating expressions with square roots of differences of squares using sine substitution
2
Solving integrals with square roots of sums of squares using tangent and secant substitution
3
Drawing and applying right triangle relationships to perform substitutions
4
Simplifying trigonometric expressions and applying u-substitution within trig integrals
Why This Matters
Trigonometric substitution is essential for solving complex integrals in calculus that can't be handled by basic techniques. You'll use this method throughout advanced calculus, differential equations, and physics to evaluate integrals involving radicals and rational functions.
Before You Start — Make Sure You Can:
This Unit Includes
3 Video lessons
Practice exercises
Learning resources
Skills
Trigonometric Substitution
Integration Techniques
Pythagorean Theorem
Right Triangles
Sine and Cosine
Tangent and Secant
U-Substitution
Calculus
BC Curriculum Aligned