CÁLCULO IV AULA 5

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 CÁLCULO IV – ENG.CIVIL – 1° SEM/2003 – AULA 5
 
Substituições Trigonométricas
 Vamos estudar agora integrais que apresentem as formas 
, 
 e 
.
 Podemos expressá-las sem o s radicais, utilizando a chamada Substituição Trigonométrica
conforme a tabela :
	Caso
	Radical
	Substit. Trigonométrica
	Transformada
	Trigonometria no Triângulo Retângulo
	I
	
	
	
	
	II
	
	
	
	
	III
	
	
	
	
 Demonstraremos o desenvolvimento do radical 
, os demais casos são análogos ...
( 
 
 
 .
Obs. : Repare que a variável final é 
. A expressão correspondente, na variável original, é 
 obtida usando-se um triângulo retângulo.
	
Exemplos :	
1 ) Achar a integral 
 
 
	
 
 
 
 ...
 ... 
 
 
.
● Devemos agora voltar à variável original “ x “ ...
 Como 
 logo 	x
 	 
 	 2	
Daí , 
,
Portanto , 
 .
2 ) 
Achar a integral 
�� EMBED Equation.3 
 
 
	
.
 
● Voltando para a variável original “ x “ ...
 
 Como 
 logo 		x
 	 
 	 	
Daí , 
,
Portanto , 
 .
3 ) 
Achar a integral 
 
 
 
 
	
 
 
 
 ...
... 
�� EMBED Equation.3 
 
Portanto ,
 
 
	 
	 
 
 
 
 
Voltando para 
 
 
	 
�� EMBED Equation.3 .
● Voltando para a variável original “ x “ ...
 
 Como 
 , 
 Logo temos ...
Daí , 
Portanto , 
 .
Exercícios :
◙ Achar as derivadas :
 1 ) 
 2 ) 
 
 3 ) 
II
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 .
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 .
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I
4
III
*
* Por Partes 
* 
x
 .
� EMBED Equation.3 ���
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2
Ver início do exercício :
� EMBED Equation.3 ���
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