# Math Help - integrals a pain

1. ## integrals a pain

here is an integral that is just driving me nuts.
Integral (cosx+sinx)/(sin2x) dx. I think that I am supposed to use the duoble angle formula for the bottom part, but im supposed to take the integral to completion, as in a final answer with +c at the end. I hope someone can help. Thanks

2. Hello, desdemoniagirl!

$\int \frac{\cos x+\sin x}{\sin2x}\,dx$
Your suspicions are correct . . .

We have: . $\frac{\cos x + \sin x}{\sin2x} \:=\:\frac{\cos x + \sin x}{2\sin x\cos x}$

Make two fractiions: / $\frac{\cos x}{2\sin x\cos x} + \frac{\sin x}{2\sin x\cos x} \;=\;\frac{1}{2\sin x} + \frac{1}{2\cos x}$

. . and we have: . $\frac{1}{2}\cdot\frac{1}{\sin x} +
\frac{1}{2}\cdot\frac{1}{\cos x} \;=\;\frac{1}{2}\csc x + \frac{1}{2}\sec x$

And now you can integrate: . $\frac{1}{2}\int(\csc x + \sec x)\,dx$ . . . . right?

3. Originally Posted by desdemoniagirl
here is an integral that is just driving me nuts.
Integral (cosx+sinx)/(sin2x) dx. I think that I am supposed to use the duoble angle formula for the bottom part, but im supposed to take the integral to completion, as in a final answer with +c at the end. I hope someone can help. Thanks
$\int \frac{\cos(x) + \sin(x)}{\sin(2x)}dx = \int \frac{\cos(x) + \sin(x)}{2\sin(x)\cos(x)}dx$

$= \int \frac{\cos(x)}{2\sin(x)\cos(x)} + \frac{\sin(x)}{2\sin(x)\cos(x)}dx$

$= \int \frac{1}{2\sin(x)} + \frac{1}{2\cos(x)}dx$

$= \int \frac{1}{2}\text{cosec}(x) + \frac{1}{2}\text{sec}(x)dx$

Hey thanks
what I was wondering is how I would work it out from there?

5. Originally Posted by desdemoniagirl
Hey thanks
what I was wondering is how I would work it out from there?
There are standard results for the integrals of those two trignometric functions:

$\int \sec(x)dx = \ln{|\sec(x) + \tan(x)|} + C$

$\int \csc(x)dx = \ln{|\csc(x) - \cot(x)|} + C$

But if you're not familiar with these and want to work it out from first principles, then have a look at these:

Integral sec(x)
Integral csc(x)

6. ## Another one

I really appreciate the help that you both gave to me earlier. Again thanks soooooooo much. I have another one or two i am still stuck on, here they are and I really hope that you can help. Thanks in advance.

Integral (cosx)/(Square root(1+sin^2(x)))

Integral (1/(x+a)(x+b))

7. Originally Posted by desdemoniagirl
I really appreciate the help that you both gave to me earlier. Again thanks soooooooo much. I have another one or two i am still stuck on, here they are and I really hope that you can help. Thanks in advance.

Integral (cosx)/(Square root(1+sin^2(x)))

Integral (1/(x+a)(x+b))