# Very Difficult Trig Sub?

n4rush0

## Homework Statement

Integral of 1/(2+sin(x)) dx

## The Attempt at a Solution

I've been told that you can use trig subs, but I never had to learn that in high school and it hasn't appeared in any of my calculus coursework.

As a side note. I've been wondering if it is possible to solve asin(x) + bcos(x) = c

Destiny153
To solve your side note use this website. it helped me out! good luck

http://www.education2000.com/demo/demo/btnchtml/sinplcos.htm [Broken]

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n4rush0
Wow, that's so cool. Thanks for the link. I'll try to remember how to derive it.

Homework Helper

## Homework Statement

Integral of 1/(2+sin(x)) dx

The standard substitution $\tan\frac{x}{2} = t$ applies for your integral. It will convert it into a integral of an algebraic function for which the method of partial fraction decomposition will get it solved.

n4rush0
Thanks, I'll try that. Is that something you just memorized or is there a certain rule that lets you know what to substitute?

Homework Helper
There are rules. That substitution will apply to an antiderivative of a function a+b\sin x/c+d\cos x and the other 3 ways of interchanging cos with sin and more generally to any algebraic function of sin and cos.

n4rush0
Where can I learn all these rules? I usually only see substitutions with x = asint, atant, or asect

Homework Helper
You're normally taught these rules of substitution in high-school. I wasn't, so I picked them up for myself from books, especially for engineers, because the proofs are missing :)

n4rush0
Okay so, given:
integral dx/(2+sinx)

tan(x/2) = t
(1/2)sec^2 (x/2) dx = dt
dx = 2cos^2 (x/2) dt

integral
2cos^2 (x/2) dt / (2+sinx)

Am I supposed to use x = arctan(2t)? If so, is it possible to simplify by drawing a triangle?

Homework Helper
Of course you have to use that. It's the whole purpose of substitution, you need to change every function of x including the dx with the approproate function of t and dt.

n4rush0
I know how to change sinx to sin 2t/sqrt(1+4t^2)
but I'm not sure how to simplify cos^2 (x/2) since it has the 1/2 in front of the x and I can't use the same trick that I used for sinx.

Homework Helper
But you need sin (2 arctan t) from the initial integral.

$$\sin (2\arctan t) = 2 (\sin\arctan t) (\cos\arctan t)$$

$$\sin\arctan t = \frac{t}{\sqrt{1+t^2}} \, , \, \cos\arctan t = \frac{1}{\sqrt{1+t^2}}$$

What about the integration element ?

Gold Member
Okay so, given:
integral dx/(2+sinx)

tan(x/2) = t
(1/2)sec^2 (x/2) dx = dt
dx = 2cos^2 (x/2) dt

integral
2cos^2 (x/2) dt / (2+sinx)

Am I supposed to use x = arctan(2t)? If so, is it possible to simplify by drawing a triangle?

It's not x=arctan(2t), but rather x=2arctan(t). that might help.

n4rush0
Thank you. I modified the integral to
dt/t^2+t+1)
Are you sure it's partial fractions?

Gold Member
There, I would actually use completing the square in the denominator, then do another trig sub.

n4rush0
Thank you. I finally get it now. I'll still have problems with the initial trig substitutions though since I'm not sure how to get tan(x/2) = t.

Unit
Letting $t = \tan(x/2)$ is part of something called a Weierstrass substitution. This is usually a pretty messy substitution, but it's good to have in your toolbox of integration tricks, especially for those pesky integrals where nothing else seems to work.