Page 225 - Calculus Workbook For Dummies

209
Chapter 11: Integration Rules for Calculus Connoisseurs

/ π 3
*j # tan i sec i d =i 14 3 - 8
2
4
5 15
/ π 4
/ π 3
1. Split off a sec i: = # tan i sec i sec i di
2
2
2
2
/ π 4 / π 3 / π 3 / π 3
2. Convert to tangents: = # tan i _ tan i + 1i sec i d =i # tan i sec i d +i # tan i sec i di
4
2
2
2
2
2
2
/ π 4 / π 4 / π 4
3. Do u-substitution with u = tani.
/ π 3
1 5 1 3 / π 3
= tan iE + tan iA
5 3 / π 4
/ π 4
1 5 1 5 1 3 1 3
= 3 - $ 1 + 3 - $ 1
5 5 3 3
14 3 8
= -
5 15
*k # tan t dt = 1 tan t - 1 tan t + 1 tan t - tant + + C
5
3
8
7
t
7 5 3
2
1. Split off a tan t and convert it to secants:
= # tan t _ sec t - 1i dt = # _ tan sec t dt - # _i tan t dt
6
2
6
2
6
t
i
2. Do the first integral with a u-substitution and repeat Step 1 with the second, then keep repeat-
ing until you get rid of all the tangents in the second integral.
1 tan t - # 4 2
7
= tan t _ sec t - 1i dt
7
1 7 1 tan t + # 2 2
5
= tan t - tan t _ sec t - 1i dt
7 5
1 7 1 5 1 3 2
= tan t - tan t + tan t - # _ sec t - 1i dt
7 5 3
1 7 1 5 1 3
t
= tan t - tan t + tan t - tant + + C
7 5 3
l # csc cot x dx = - 2 csc / 5 2 x + 2 csc / 1 2 x + C
3
x
5
1. Split off csc cot x: = # csc - / 1 2 x cot x csc cot x dx.
x
2
x
2. Convert the even number of cotangents to cosecants with the Pythagorean Identity.
= # csc - / 1 2 x _ csc x - 1i csc cot x dx
2
x
3. Finish with a u-substitution.
= # csc x csc cot x dx - # csc - / 1 2 x csc cot x dx
/ 3 2
x
x
= u # / 3 2 ^ - du - # u - / 1 2 ^ - duh
h
2 / 5 2 / 1 2
= - u + 2 u + C
5
2 / 5 2 / 1 2
= - csc x + 2 csc x + C
5
*m # x 9 + 16 x dx = 1 _ 9 + 16 x i / 3 2 + C
2
2
48
1. Rewrite as # x ^ 4 h 2 3 2 dx.
x +
u
2. Draw your SohCahToa triangle where tan =i a . See the following figure.