# What`s the integral of this?

1. Dec 30, 2008

### Penultimate

How can i solve the integral of 1/(sinx + cosx)?

2. Dec 30, 2008

### Dick

I would start by multiplying numerator and denominator by (sin(x)-cos(x)). Does that give you any ideas?

3. Dec 30, 2008

### rootX

General technique:

1/(sinx + cosx)

Use: t = tan (1/2.x)
and find sin(x), cos(x) in terms of t {Using a traingle helps}.

This solves all kind of problems where you ve sin,cos in denominators and your book should have also discussed this somewhere.

You should first prove that sin(t) = 2x / (1+x^2) and cos(t) = (1-x^2)/(1+x^2)
and dt = 2/(1+x^2).dx

Last edited: Dec 30, 2008
4. Jan 1, 2009

### Penultimate

So the general tecnique is to have t= tan 1/2 of the variable?

5. Jan 1, 2009

### Dick

It's a general way to replace sin(x), cos(x) and dx with rational functions (quotients of polynomials) of another variable. It should work. It's not necessarily the easiest way to do any given integral.

6. Jan 1, 2009

### rootX

Just use it when you can't think of anything else. Mostly likely, it becomes a problem of partial fractions and inverse of sines,cosines,tan (integration table always have their integration formulas),.. it's just messing up algebra/using wrong trigonometry relations that make this harder to use but you always know what to do in the next step.

There are too many ways to solve these problems. Here's another way:
convert sin (x) + cos(x) = A.sin(x+phi) .. and you will have 1/A*sin (...), and now you can use integration table.

7. Jan 1, 2009

### HallsofIvy

Staff Emeritus
And don't forget what Dick originally suggested. Multiplying both numerator and denominator of 1/(sin(x)+ cos(x)) by sin(x)- cos(x) you have (sin(x)- cos(x))/(sin2(x)- cos2(x))= sin(x)/(sin2(x)- cos2(x))- cos(x)/(sin2(x)- cos2(x))= sin(x)/(sin2-(1- sin2(x))- cos(x)/((1- cos2(x))- cos2(x))= sin(x)/(2sin2-1)- cos(x)/(1- 2cos2(x)).