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

= + +