7-3_1 - Techniques of Integration Techniques of Integration...

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Unformatted text preview: Techniques of Integration Techniques of Integration I Integration Formulas I Substitution I trigonometric substitution I Integration by Parts Objective : To “get rid of” the square root in an integral with a factor of the form p a 2 − x 2 or 1 √ a 2 − x 2 Objective : To “get rid of” the square root in an integral with a factor of the form p a 2 − x 2 or 1 √ a 2 − x 2 or p a 2 + x 2 or 1 √ a 2 + x 2 ( also works for 1 a 2 + x 2 ) Objective : To “get rid of” the square root in an integral with a factor of the form p a 2 − x 2 or 1 √ a 2 − x 2 or p a 2 + x 2 or 1 √ a 2 + x 2 ( also works for 1 a 2 + x 2 ) or p x 2 − a 2 or 1 √ x 2 − a 2 Tools : 1. Substitution 2. Trigonometric identities 3. In a right triangle Type 1 : Sine substitutions p a 2 − x 2 or 1 √ a 2 − x 2 , a > Type 1 : Sine substitutions p a 2 − x 2 or 1 √ a 2 − x 2 , a > Substitution : x = a sin( θ ) ⇒ dx = a cos( θ ) d θ, − π 2 ≤ θ ≤ π 2 Type 1 : Sine substitutions p a 2 − x 2 or 1 √ a 2 − x 2 , a > Substitution : x = a sin( θ ) ⇒ dx = a cos( θ ) d θ, − π 2 ≤ θ ≤ π 2 Why? Type 1 : Sine substitutions p a 2 − x 2 or 1 √ a 2 − x 2 , a > Substitution : x = a sin( θ ) ⇒ dx = a cos( θ ) d θ, − π 2 ≤ θ ≤ π 2 Why? Simplification : p a 2 − x 2 = q a 2 − a 2 sin 2 ( θ ) = q a 2 (1 − sin 2 ( θ )) = q a 2 cos 2 ( θ ) = a cos( θ ) Sometimes, it is helpful to draw a right triangle. Sometimes, it is helpful to draw a right triangle. In this right triangle, sin( θ ) = x a Sometimes, it is helpful to draw a right triangle. In this right triangle, sin( θ ) = x a cos( θ ) = √ a 2 − x 2 a Sometimes, it is helpful to draw a right triangle. In this right triangle, sin( θ ) = x a cos( θ ) = √ a 2 − x 2 a tan( θ ) = x √ a 2 − x 2 Sometimes, it is helpful to draw a right triangle. In this right triangle, sin( θ ) = x a cos( θ ) = √ a 2 − x 2 a tan( θ ) = x √ a 2 − x 2 cot( θ ) = √ a 2 − x 2 x Sometimes, it is helpful to draw a right triangle....
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This note was uploaded on 04/23/2011 for the course MATH 022 taught by Professor Salathe during the Spring '09 term at Lehigh University .

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7-3_1 - Techniques of Integration Techniques of Integration...

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