Question

trig substitution

Answer 1

for step 2 I am having trouble please help

Answer 2

x = (7/3) sec t
x^2 = ...?

9 x^2 =... ?

Answer 3

find dt also,

x = (7/3) sec t
differentiate this w.r.t t

Answer 4

ok i got 49sec^2(t) for 9x^2

Answer 5

thats correct,
now dt

Answer 6

(7/3)tan(t)sec(t)

the format of the question was int f(t)dt

i thought that i wouldn't combine dt and f(t)

Answer 7

because it asked for f(t)=

Answer 8

yes,
x = (7/3) sec t
when you differentiate this,
you get some function in 't' times dt
now this function in 't' will go with f(t)
hence we need to find dx

Answer 9

x= (7/3) sec t
differentiate this w.r.t t
what do u get ?

Answer 10

idk what wrtt means

Answer 11

with respect to

Answer 12

find dx/dt

Answer 13

ok so dt=(7/3)tan(t)sec(t)

x'(7/3)tan(t)sec(t)

Answer 14

could you help me out with this part?

Answer 15

dx/dt = (7/3) tan(t)sec(t)

right ?
so
dx = (7/3) tan(t)sec(t) dt
got this ?

Answer 16

ok am i finding the first order differential equation?

Answer 17

for dx = (7/3) tan(t)sec(t) dt

Answer 18

or do i just take the derivative of (7/3)tan(t)sec(t) again to get -7/6(cos(2 t)-3)sec^3(t)

Answer 19

plug this in your original integral

Answer 20

not again

Answer 21

plug in these :

dx = (7/3) tan(t)sec(t) dt
and x= (7/3) sec t
in your original integral

Answer 22

try to simplify the denominator

Answer 23

i think you know what \(\sec^2 x-1 = .. \)

Answer 24

oh i see where the dx was coming from.

ok so cos^2(x)

Answer 25

where does cos^2 x come from ?

Answer 26

i mean sqrt(cos^2(x))

does sqrt sec^2(x)-1=sqrt cos^2(x)

Answer 27

do u know about the famous pythagorean identity.
\(\Large \sec^2 x = 1+\tan^2 x\)
?

Answer 28

lol well now i do.

so for the denominator 7/3sqrt(1-tan^2x)

Answer 29

i mean 1+tan^2(x)

Answer 30

umm..no, see
\(\Large \sqrt{9x^2-49} = \sqrt{49\sec^2 t-49} \\ \Large = \sqrt {49}\sqrt {\sec^2-1} = 7 \sqrt{\tan ^2 x} = 7\tan x\)

see if you get those steps

Answer 31

i mean 7 tan t

Answer 32

ok that made a lot more sense now we simiplify the numerator and denominator

Answer 33

yes, something getting cancelled ?

Answer 34

yeah the 7 and tan

Answer 35

so sec(t)/3

Answer 36

yep, you are correct :)