How could this Integration be solved? Any possible Steps.

Question

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

Any steps leading to a simple answer would be really appreciated. Thanks !

@ErikMiehling can you show a few steps? I am really poor at integration but I need to solve it partially ti impress my boss! — Jishan, Mar 20 '14 at 17:27
You can split the fraction if it helps: $\frac{1}{x} + \frac{sinx-1}{x+cosx}$ (And it DOES help) — imranfat, Mar 20 '14 at 17:36

score 4 · Accepted Answer · edited Mar 20 '14 at 17:52

4

$\displaystyle\int \dfrac{\cos x + x \sin x}{x(x + \cos x)}dx$

$\displaystyle = \int \dfrac{x+\cos x}{x(x+\cos x)}dx + \int \dfrac{x\sin x - x}{x(x+\cos x)}dx$

$\displaystyle= \ln|x| - \int \frac{d(x + \cos x)}{x+\cos x}dx = \ln \left|\frac x{x+\cos x}\right| + c$.

edited Mar 20 '14 at 17:52

lab bhattacharjee

answered Mar 20 '14 at 17:34

r9m

score 2 · Answer 2 · answered Mar 20 '14 at 19:07

2

$$\ln\left|\frac{x}{x+\cos x}\right|$$

Just a minute and I'll bring you the proof.

answered Mar 20 '14 at 19:07

sidneimv

2 Answers2