Recently, I've been facing a no. of problems that involve integral of the type: $\int f(x)dx/f'(x)$ and also of the type $f(x)/√f'(x)$. So, is there any special method for such type of questions? One such question was $$\int (x+sinx) \frac1{cosx+1} $$ Clearly this is of the first type. I could make out that such questions must involve the use of integration by parts but I'm unable to apply that. It makes the problem more complex.
Any help is appreciated.
If you want to use integration by parts, you'll get:
$$\int y(x)z'(x)\space\text{d}x=y(x)z(x)-\int y'(x)z(x)\space\text{d}x$$
So, set $y(x)=f(x)$ and $z'(x)=\frac{1}{\sqrt{f'(x)}}$:
$$\int\frac{f(x)}{\sqrt{f'(x)}}\space\text{d}x=f(x)\cdot\int\frac{1}{\sqrt{f'(x)}}\space\text{d}x-\int\left[f'(x)\cdot\int\frac{1}{\sqrt{f'(x)}}\space\text{d}x\right]\space\text{d}x$$
Now, notice for $\frac{x+\sin(x)}{1+\cos(x)}$ that you cannot use (1) because:
$$\frac{x+\sin(x)}{1+\cos(x)}=\frac{x+\sin(x)}{\frac{\text{d}}{\text{d}x}\left(x+\sin(x)\right)}\ne\frac{\frac{\text{d}}{\text{d}x}\left(x+\sin(x)\right)}{x+\sin(x)}$$