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Trigonometric Substitutions |
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Trigonometric Substitutions
Trig. substitutions are usually used for integrals that
involve square roots of quadratic expressions; the trig
substitution is used to eliminate the radical sign; the following
substitutions are made:
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Term involve in the Integral
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Substitution
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Trig. Identity
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u = a sin q |
cos2 q = 1 - sin2 q |
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u = a tan q |
sec2 q = 1 + tan2 q |
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u = a sec q |
tan2 q = sec q -1 |
where a is any real number and u is any
variable.
ex.
Substitute: x = sec q
dx = sec q tan q dq
Substitute:
u = sin q
du = cos q dq
The solution must be in terms of x:
From the substitution of x = sec q,
the diagram of a right triangle gives;
The final solution in terms of x is:
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