Trigonometric Substitution | Techniques of Integration

Trigonometric substitution is employed to integrate expressions involving functions of (a2 − u2), (a2 + u2), and (u2 − a2) where "a" is a constant and "u" is any algebraic function. Substitutions convert the respective functions to expressions in terms of trigonometric functions. The substitution is more useful but not limited to functions involving radicals.
 

Use the following suggestions:

When the integrand involves...

  • (a2 − u2), try u = a sin θ
  • (a2 + u2), try u = a tan θ
  • (u2 − a2), try u = a sec θ

The substitution may be represented geometrically by constructing a right triangle.
 

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