Enter expression form and parameter to find the appropriate trigonometric substitution
Expression Form
Parameter a
Result
Recommended Substitution
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Simplified Form
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Step-by-Step Derivation
Trigonometric Substitution Rules
sqrt(a^2 - x^2): x = a sin(theta), dx = a cos(theta) dtheta
Result: sqrt(a^2 - x^2) = a cos(theta)
sqrt(a^2 + x^2): x = a tan(theta), dx = a sec^2(theta) dtheta
Result: sqrt(a^2 + x^2) = a sec(theta)
sqrt(x^2 - a^2): x = a sec(theta), dx = a sec(theta) tan(theta) dtheta
Result: sqrt(x^2 - a^2) = a tan(theta)
Trigonometric substitution is a powerful integration technique from calculus. It leverages Pythagorean identities to simplify expressions with square roots of quadratic forms.
⚠After integration, use a reference triangle to convert back from theta to x. The triangle sides depend on the substitution used.
What Is Trigonometric Substitution?
Trigonometric substitution replaces a variable with a trigonometric function to simplify square root expressions. It is a key technique in calculus integration, especially for integrals involving quadratic forms under square roots.
Type 1: sin
sqrt(a^2-x^2): x = a sin(theta). Uses 1-sin^2 = cos^2. Result: a cos(theta). Best for circle-related integrals.
Type 2: tan
sqrt(a^2+x^2): x = a tan(theta). Uses 1+tan^2 = sec^2. Result: a sec(theta). Best for hyperbola-related integrals.
Type 3: sec
sqrt(x^2-a^2): x = a sec(theta). Uses sec^2-1 = tan^2. Result: a tan(theta). Best for integrals with vertical asymptotes.
Back Substitution
Draw a right triangle: sides correspond to a, x, and sqrt expression. Use it to find trig values of theta in x.
Teaching Example: Integral of sqrt(9-x^2) dx. Form: sqrt(a^2-x^2) with a=3. Substitution: x = 3 sin(theta), dx = 3 cos(theta) dtheta. sqrt(9-x^2) = 3 cos(theta). Integral = integral of 9 cos^2(theta) dtheta = (9/2)(theta + sin(theta)cos(theta)) + C. Then convert back.
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