Question

How do I start to get the integral of dx/1+sinx ? Do I multiply with 1-sinx/ 1-sinx ?

Answer 1

do u know this standard substitution?
t= tan (x/2)
?

Answer 2

It is in the the textbook, but I really struggle with it.

Answer 3

can u find sin x in terms of t
with t= tan (x/2) ??

Answer 4

or do u know the formula of sin 2x in terms of tan x ??

Answer 5

let me tell u
sin2x = 2 tan x /(1+ tan^2 x)
so
sin x = 2tan (x/2)/(1+ tan ^2 (x/2)) = 2t/(1+t^2)
and since
x= 2 tan^-1 t
dx = 2/(1+t^2)
clear till here ?

Answer 6

i think @Iventer 's method will work too...

Answer 7

i want to write it down to check it. i only have sin2x = 2sinxcosx

Answer 8

how do you get to the tan formula

Answer 9

its standard , remember it.
and yes , inverter's way will also work, but i want inverter to know this substitution as it helps in many different problems.

Answer 10

i've never used half-angle theorems in integrals though

Answer 11

all i know is the formula for \[\sin (\alpha \pm \beta)\]
\[\cos (\alpha \pm \beta)\]
then the pythagorean identities

and it has kept me happy for a long time

Answer 12

if t= tan (x/2)
cos x = (1-t^2)/(1+t^2)
sin x = (2t)/(1+t^2)
dx = (2)/ ( 1+t^2)
note it down.

Answer 13

i suggest you memorize as little as possible @Iventer and just practice and learn to manipulate your surroundings to your advantage

Answer 14

im not saying dont listen to hartnn though

Answer 15

ok i have that @hartnn

Answer 16

now substitute dx and sin x in your integral , you should get 2/((1+t)^2)

Answer 17

give me second

Answer 18

ok i have it. what next

Answer 19

can u integrate that ?

Answer 20

what answer u have in your answer key?

Answer 21

is it in half angle form ??

Answer 22

else this entire procedure will be more difficult than to multiply and divide by (1-sin x)

Answer 23

we dont really work with the half angle. i am really lost as i am studying long distance and have to figure all this out on my own with just a textbook as help. I have to know the tan(x/2) method as well, but i think i will have to look at it more. I can use what u have given me. but i need more time. i will close for now but get back later. thank you so much for your help so far xxxxx

Answer 24

ok,going with your original question,YES, multiply and divide by (1-sin x)
then u get
(1-sin^2 x) = cos^2 x in the denominator,
then just separate out the denominators, and u will get
1/(cos ^2 x) - sinx/ cos^2 x = sec^2 x - sec x tan x
now this is easy to integrate, isn't it ?

Answer 25

much better, yes thank u lots

Answer 26

there are many approaches to solve a particular integral, here it was simpler to use (1-sin x).
But in problems like integral of {1/(a sinx + b cos x +c)} dx,
that substitution will be simplest to solve....maybe in near future, u have to deal with such integrals.

Answer 27

i have one question that is asking to solve that way, so yes i have to look at it in near future. but i have idea where to start now with sin, cos and dx u gave me. much appreciated