Chapter 7 Advanced intigration

Question 1

integration by parts

Answer 1

this is a way to integrate products
begin by assigning values as u and dv
then find du and v

from here set it as uv -int(vdu)

Question 2

when you integrate by parts and get a loop

Answer 2

integrate by parts until you get the original integral. and then use addition to isolate it

treat it like a variable

Question 3

integral of lnx

Answer 3

xlnx -x

Question 4

trig integrals integration

Answer 4

use various substitutions to integrate advanced trig functions

Question 5

if sin is odd and positive

Answer 5

isolate one sinx and turn the rest into cosines

then u sub for cosine

Question 6

if cos is odd and positive

Answer 6

keep one cosx and then turn the rest into sins.

then usub for sinx

Question 7

sin and cos identity

Answer 7

sin^2 +cos^2=1

Question 8

tan and sec identity

Answer 8

tan^1 +1 = sec^2

Question 9

if cos and sin are both even

Answer 9

use half angle identities on them to turn it into powers of cos2x

then use the odd and even strategies until an integral is possible

Question 10

if tangent is odd and more than 1

Answer 10

leave one and turn the rest into secants. from then u sub with sec

Question 11

if sec is even

Answer 11

then use u sub with tan

Question 12

sin^2x half angle

Answer 12

1/2(1-cos2x)

Question 13

cos^2x half angle

Answer 13

1/2(1+cos2x)

Question 14

sin(2x) double angle

Answer 14

2sinxcosx

Question 15

cos(2x)=

Answer 15

cos^2-sin^2

Question 16

int tanx

Answer 16

ln|sec|+c

Question 17

int secX

Answer 17

ln|secx+tanx| +c

Question 18

int cotx

Answer 18

ln|sinx|+c

Question 19

int cscx

Answer 19

-ln|cscx +cotx|+c

Question 20

if tan is even and sec is odd

Answer 20

turn the tan into secants and then multiply. from then gl

Question 21

trig substitutiuon

Answer 21

an integral method when you use the triangle

Question 22

a^2-x^2

Answer 22

use the sub asinx

Question 23

a^2+x^2

Answer 23

atanx

Question 24

x^2-a^2

Answer 24

asecx

Question 25

fracion decompositon

Answer 25

breaking up the frac over the numerator

Question 26

lim (x to inf) e^x

Answer 26

inf

Question 27

lim (x to -inf) e^x

Answer 27

0

Question 28

lim (x to inf) e^-x

Answer 28

0

Question 29

lim (x to -inf) e^-x

Answer 29

inf

Question 30

lim (x to inf) ln x

Answer 30

inf

Question 31

lim (x to 0+)

Answer 31

-inf

Question 32

lopitals o/o and inf/inf

Answer 32

1. derive the top and bottom separately

2. take the limit of the top over the limit of the bottom

Question 33

lopitals 0*inf

Answer 33

1. make it into a fraction by placing one of the sides on the bottom to the -1 power.
2. then treat like a normal lopitals

Question 34

lopitals 0^inf inf^0 etc

Answer 34

1. take the ln of BOTH sides
2. use log rules to make it multiplication
3. use the multiplication to find the limit.
4. undo the log on the other side by taking e^ what you get

Question 35

how to integrate when the numerator is greater than the denom

Answer 35

long devision and then go from there

Question 36

if we have soemthing over the square root of a fraction that can not be simplified

Answer 36

use completing the square adn tehn go from there

Question 37

what to do if we have something like x^2/x^2 +1

Answer 37

change the top to x^2 +1-1 and then split up

you would get one part to cancel out and then from there you could integrate easily the other parts.

Question 38

what if we have somethig like dx/1+cosx

Answer 38

we need to rationalize the denom. for instance this means that we need to multiply both sides by (1-cosx)/(1-cosx)