00:01
In this problem, use the substitution t is equal to tan x by 2 to transform the integrant into a rational function of t and then evaluate the integral.
00:13
So remember to use absolute values where appropriate.
00:18
So now what happens is you have to find out integration of 3 by 3 sin x minus 4 cos x dx is equal to 0.
00:31
So we will try to calculate it in the form of t that is equal to tan x by 2.
00:41
So this is what it is said that you have to use the integrant right into a rational function of t right.
00:53
So what is the relation between sin x and tan x by 2? so this is the relation basically.
01:00
Sin x is equal to 2 tan x by 2 whole divided by 1 plus tan square x by 2 right.
01:08
So now this is equal to 2t by 1 plus t square right.
01:13
Now for similarly for cos x if you say so for cos x it would be 1 minus tan square x by 2 upon 1 plus tan square x by 2.
01:30
So now this will be 1 minus t square whole upon 1 plus t square.
01:36
So this is the formula you have to remember in the form of tan x by 2.
01:41
Now as we know t is equal to tan x by 2 right.
01:45
So if you differentiate this equation dt by dx.
01:49
So differentiation of tan x by 2 would be sec square x by 2 right.
01:54
But x will be also differentiated as per the chain rule that is half right.
02:00
Or you can also write it as half 1 multiplied by whole multiplied by 1 plus tan square x by 2 yes.
02:10
So this is an identity in which 1 plus tan square x is nothing but which is equal to sec square x.
02:17
So can you replace it sec square x by 2 by 1 plus tan square x by 2 yes.
02:23
So it becomes half multiplied by 1 plus whole multiplied by you can say 1 plus t square right.
02:33
So now what it becomes? it becomes 2 dt 2 dt by 2 dt by 1 plus t square right...