trig_substitution

trig_substitution - Trigonometric
Subs0tu0on
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Unformatted text preview: Trigonometric
Subs0tu0on
 Sec0on
8.3
 Integrals
involving





 2 − x 2 a θ a x a 2 − x 2 = a cos(θ ) θ a x = a sin(θ ) € 



Subs0tu0on:
 x = a sin(θ ) dx = a cos(θ ) dθ € € € 



Back‐subs0tu0on:
 €θ = sin−1 ( x / a) x sin(θ ) = a cos(θ ) = a2 − x 2 a € a 2 − x 2 = a cos(θ ) Integrals
involving





 2 + x 2 a θ a θ x a = a 2 + x 2 cos(θ ) a2 + x 2 € 



Subs0tu0on:
 € x = a tan(θ ) dx = a sec 2 (θ ) dθ a 2 + x 2 = a sec(θ ) € € x = a tan(θ ) 



Back‐subs0tu0on:
 −1 € θ = tan ( x / a) x sin(θ ) = a2 + x 2 a cos(θ ) = a2 + x 2 Integrals
involving





 2 − a 2 x a € θ θ x a x = a sec(θ ) € € 



Subs0tu0on:
 x = a sec(θ ) dx = a sec(θ ) tan(θ ) dθ x 2 − a 2 = a tan(θ ) € € x 2 − a 2 = a tan(θ ) € € € 



Back‐subs0tu0on:
 −1 x θ = sec ( x / a) ⇒ sec(θ ) = a sin(θ ) = tan(θ ) = x 2 − a2 a , cos(θ ) = x x x 2 − a2 a ...
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