is equal to –

The correct answer is: In + C

let I be the integral ∫
let  √ (2xe^sinx +1) be t
On differentiating both sides with respect to x, we get
2x(e^sinx) cos x + e^sinx (2) =dt/dx
2e^sinx[ xcosx +1] =dt/dx
Multiplying and dividing by x, we get
2xe^sinx[xcosx+1]/x =dt/dx
Or
(t-1) [xcosx+1]/x =dt/dx
[xcosx+1]/x = dt/(t-1)dx
Using this in the integral , we get

now, put t=z²
this gives I=  
using partial fraction, we get
I = 

I = In + C

substitution method of integration helps us reduce the integral to simpler terms.