I tried to simplify the equation to -1 +2/(1-sin(x))
#How do I integrate this?
swift nest
I tried to simplify the equation to -1 +2/(1-sin(x))
I think u can use the sub u=sinx and then integrate (1+u)/(1-u) by parts
Actually wait no
$$\frac{1+sinx}{1-sinx} = -\frac{1+sinx}{sinx-1} = -\frac{(sinx-1)+2}{sinx-1} = -1-\frac{2}{sinx-1}$$
half baneBOT
Arizus
shut anvil
Can you not multiply the top and bottom by (1+sinx)?
Resulting in (1+sinx)^2/cos^2x
Which can be simplified into (1+2sinx+sin^2x)/cos^2x
Further simplified into sec^2x+2tanxsecx+tan^2x
then use trig IDs to write 2sec^2x+2tanxsecx+x
Then integrate from there
Wait is my method wrong?
swift nest
The sub probably involves tan
Cuz i think you can get a sec² and a tan and then do it by parts
Ah i think i see
sinx-1
= 2sin(x/2)cos(x/2)-sin²(x/2)-cos²(x/2)
= 2tan(x/2)/sec²(x/2)-1/sec²(x/2)-tan²(x/2)/sec²(x/2)
= -(tan²(x/2) -2tan(x/2)+1)/sec²(x/2)
= -(tan(x/2)-1)²/sec²(x/2)
So -2/(sinx-1) = sec²(x/2)/(tan(x/2)-1)²
Then u can use the sub u = tan(x/2)-1
And everything cancels cuz du/dx = sec²(x/2)/2 so dx = 2/sec²(x/2) and then the sec²(x/2) cancels
So u just get an integral of 1/u²
midnight lion
whered the x/2 come from
swift nest
whered the x/2 come from
The identity that sinx = 2sin(x/2)cos(x/2)
dim cove