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Trigonometric Substitutions

Main formula:

1. For , substitute .

2. For , substitute .

3. For , substitute .

Hints and tips:

• Remember to make the substitution as well as the main substitution.

• You can sometimes use trigonometric substitution even when you don’t have a square root. However, in these cases, you should only be using the tangent substitution. If you get one of the other ones, then you should have been able to factor the algebraic expression, and trigonometric substitution was not necessary.

• If you have a linear term (x or u, as opposed to x² or u²), complete the square and make a substitution to eliminate the linear term before you make the trigonometric substitution.

• If the coefficient of the quadratic term (x²) is negative, factor the negative out of all terms before completing the square.

• After you make the trigonometric substitution, it becomes a trigonometric integral that you can handle using the techniques in the previous section.

Trigonometric Substitutions

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