Download - Supplementary Material Trigosub
-
University of the Philippines Los Baos College of Engineering and Agro-industrial Technology
Department of Engineering Science
ENSC 21 - SUPPLEMENTARY MATERIAL COPYRIGHT 2015 BY JSVENTURA
Supplementary Material (Week 1) Trigonometric Substitution Table of Trigonometric Substitutions
Expression Substitution Identity = sin , 2 2 1 = + = tan , 2 < < 2 1 + = = sec , 0 < 2 < 32 1 =
Sine substitution Tangent substitution Secant substitution
sin =
tan =
sec =
Example:
1. Evaluate
.
Solution: Let = 3 sin , = 3 cos 9 = 9 9 = 9 = 3|cos| = 3 cos Thus, we have
9
= 3 cos 9 3 cos = = = ( 1)
= cot + =
+
2. Find
.
Solution: Let = 2 tan , = 2
-
University of the Philippines Los Baos College of Engineering and Agro-industrial Technology
Department of Engineering Science
ENSC 21 - SUPPLEMENTARY MATERIAL COPYRIGHT 2015 BY JSVENTURA
+ 4 = 4( + 1) = 4 = 2|sec| = 2 sec Thus, we have
1
+ 4 = 2 4 2 sec = 14 sec
=
=
Let = sin
1
+ 4 = 14 cos = 14 = 14 1 + = 14 sin + = csc 4 + = +
+
3. Evaluate
, where > 0.
Solution: Let = sec , = sec tan = ( 1) = = |tan| = tan Therefore,
= sec tan
tan = sec = |sec + tan| +
=
(see figure above)
=
+
+ = + ln +
= ln
= + +