integrate sinx dx / (1+sinx)^1/2

Expert Answers
e-devam eNotes educator| Certified Educator

`intsinx/sqrt(1+sinx)dx `

Let, `u=sinx, cosx=sqrt(1-u^2)` (draw a right triangle to get support for this value of cosx)



The integral then takes the form,

`intu/(sqrt(1+u)sqrt(1-u^2))du `

=`intu/(sqrt((1+u)(1-u)(1+u)))du `


Make a second substitution as `sqrt(1-u)=t `



The integral now becomes

`int((1-t^2)(-2t))/((2-t^2)t)dt `




Integration by decomposing into partial fraction yields


Putting back the value of t,


Finally, putting back the value of u gives,

`=-1/2(4sqrt(1-sinx)+sqrt2log(sqrt2-sqrt(1-sinx))-sqrt2log(sqrt2+sqrt(1+sinx)))+C` Where, C is the constant of integration.