1. ## indefinite integral

I seem to be getting worse and worse at integrating haha.. I can get this in a form that looks simple but for the life of me cannot seem to get past it!!

$\displaystyle \int \frac{sin(kx)}{\sqrt{2+cos(kx)}}dx$ where k does not equal zero

let $\displaystyle cos(kx) = u$ therefore $\displaystyle dx = \frac{du}{-ksin(kx)}$

so it becomes $\displaystyle -\frac{1}{k} \int \frac{du}{\sqrt{2 + u}}$

for some reason I cannot get past here.. Very frustrating!!

2. Originally Posted by U-God
I seem to be getting worse and worse at integrating haha.. I can get this in a form that looks simple but for the life of me cannot seem to get past it!!

$\displaystyle \int \frac{sin(kx)}{\sqrt{2+cos(kx)}}dx$ where k does not equal zero

let $\displaystyle cos(kx) = u$ therefore $\displaystyle dx = \frac{du}{-ksin(kx)}$

so it becomes $\displaystyle -\frac{1}{k} \int \frac{du}{\sqrt{2 + u}}$

for some reason I cannot get past here.. Very frustrating!!
Try this substitution [for the original integral]:

Let $\displaystyle u=\cos(kx){\color{red}+2}$

See how you go with this one

--Chris

3. Originally Posted by Chris L T521
Try this substitution [for the original integral]:

Let $\displaystyle u=\cos(kx){\color{red}+2}$

See how you go with this one

--Chris
you're a genius
thanks tons

4. Actually one more while we're at it would be fantastic!

$\displaystyle \int \frac{1}{5x - \sqrt{x}}dx$
I don't really know how to begin this one, I tried looking for a substitution but couldn't see one that work (although I may be blind).

Cheers,

5. Originally Posted by U-God
Actually one more while we're at it would be fantastic!

$\displaystyle \int \frac{1}{5x - \sqrt{x}}dx$
I don't really know how to begin this one, I tried looking for a substitution but couldn't see one that work (although I may be blind).

Cheers,
Note that $\displaystyle \frac{1}{5x-\sqrt{x}}=\frac{1}{\sqrt{x}\left(5\sqrt{x}-1\right)}$

Then make the substitution $\displaystyle u=5\sqrt{x}-1$

See how you go with this one

--Chris

6. I can never see things like you do!! Cheers!

7. Okay, I got one here that I thought I'd done right but it doesn't match the answer:

$\displaystyle \int \sqrt{1 + 4x^2} dx$ so I let $\displaystyle x = \frac{1}{2}sinh(u)$
therefore $\displaystyle dx = \frac{1}{2}cosh(u)du$

subbed it in: $\displaystyle \frac{1}{2}\int cosh^2(u)du$

used identity to re-arrange: $\displaystyle \frac{1}{4} \int cosh(2u)du + \frac{1}{4}\int du$

evaluated: $\displaystyle \frac{1}{8}sinh(u) + \frac{1}{4}u + C$

re arranged the substitution I made earlier for $\displaystyle u = arcsinh(2x)$
subbed u back in yielded $\displaystyle \frac{1}{8}sinh(arcsinh(2x)) + \frac{1}{4}arcsinh(2x) + C$

which cancelled to give $\displaystyle \frac{1}{4}x + \frac{1}{4} arcsinh(2x) + C$

however the answer I'm given is: $\displaystyle \frac{1}{4}(arcsinh(2x) + 2x\sqrt{1 + 4x^2}) + C$

Hopefully someone can spot what I've done wrong. Thanks!

8. Originally Posted by U-God
Okay, I got one here that I thought I'd done right but it doesn't match the answer:

$\displaystyle \int \sqrt{1 + 4x^2} dx$ so I let $\displaystyle x = \frac{1}{2}sinh(u)$
therefore $\displaystyle dx = \frac{1}{2}cosh(u)du$

subbed it in: $\displaystyle \frac{1}{2}\int cosh^2(u)du$

used identity to re-arrange: $\displaystyle \frac{1}{4} \int cosh(2u)du + \frac{1}{4}\int du$

evaluated: $\displaystyle \frac{1}{8}sinh(2u) + \frac{1}{4}u + C$ Mr F added the red 2.

re arranged the substitution I made earlier for $\displaystyle u = arcsinh(2x)$
subbed u back in yielded $\displaystyle \frac{1}{8}sinh(arcsinh(2x)) + \frac{1}{4}arcsinh(2x) + C$

which cancelled to give $\displaystyle \frac{1}{4}x + \frac{1}{4} arcsinh(2x) + C$

however the answer I'm given is: $\displaystyle \frac{1}{4}(arcsinh(2x) + 2x\sqrt{1 + 4x^2}) + C$

Hopefully someone can spot what I've done wrong. Thanks!