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Calculus II practice

Trigonometric Substitution Integrals worksheets

Use the three standard radical substitutions with domain-controlled simplification and explicit back-substitution.

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State the substitution and angle domain, transform dx and any bounds, then back-substitute to x.

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Each sheet asks students to expose the identity, substitution, decomposition, or limiting process that makes the integral valid. Coefficients and scales vary by seed while the selected families stay balanced.

Learning objective

What this activity builds

Match each quadratic radical with sine, tangent, or secant substitution and return an exact result in x.

Worked example

Which substitution matches sqrt(a^2-x^2)?

x = a sin(theta), with theta in [-pi/2, pi/2].

Then the radical is a cos(theta), and cos(theta) is nonnegative on the stated interval.

Teaching notes

Make the practice count

  • Require an angle domain before simplifying a square root.
  • Transform definite bounds instead of back-substituting them.
  • Label the reference triangle.

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Math SheetNameScoreTeacherDateTrigonometric Substitution IntegralsChoose the standard substitution, state its domain, simplify the radical, and back-substitute to x. Give exact values for definite integrals.1.Evaluate exactly: integral sqrt(x^2 - 9) dx. Use x = 3 sec(theta); x >= 3; 0 <= theta < pi/2.2.Evaluate exactly: integral from 0 to 3 2 sqrt(9 + x^2) dx. Use x = 3 tan(theta); all real x; -pi/2 < theta < pi/2.Show the limit, decomposition, identity, or substitution that justifies each result.math-sheet.comPage 1 of 3
Math SheetNameScoreTeacherDateTrigonometric Substitution IntegralsChoose the standard substitution, state its domain, simplify the radical, and back-substitute to x. Give exact values for definite integrals.3.Evaluate exactly: integral sqrt(16 - x^2) dx. Use x = 4 sin(theta); |x| <= 4; -pi/2 <= theta <= pi/2.4.Evaluate exactly: integral from 0 to 3 2 sqrt(9 - x^2) dx. Use x = 3 sin(theta); |x| <= 3; -pi/2 <= theta <= pi/2.Show the limit, decomposition, identity, or substitution that justifies each result.math-sheet.comPage 2 of 3
Math SheetNameScoreTeacherDateTrigonometric Substitution IntegralsChoose the standard substitution, state its domain, simplify the radical, and back-substitute to x. Give exact values for definite integrals.5.Evaluate exactly: integral 3 sqrt(64 + x^2) dx. Use x = 8 tan(theta); all real x; -pi/2 < theta < pi/2.6.Evaluate exactly: integral from 8 to 8sqrt(2) 3 sqrt(x^2 - 64) dx. Use x = 8 sec(theta); x >= 8; 0 <= theta < pi/2.Show the limit, decomposition, identity, or substitution that justifies each result.math-sheet.comPage 3 of 3

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