What is the integral of 1 plus sin2xcscx?

Answer 1

The indefinite integral with respect to x (assumed that, for future reference you should always list the variable of integration (dx, dy, dz, dq, dt, etc...)) of:

f(x)=1+sin2(x)csc(x)

csc(x) is the same as 1/sin(x), and sin2(x)/sin(x)=sin(x), so f(x) can be simplified to:

f(x)=1+sin(x)

the antiderivatives of both of the components of this function are easy and known. Since they are added together and not multiplied or divided, they can be and integrated individually and then added together in the end (capital-letter function names denote antiderivatives):

f(x)=h1(x)+h2(x)

h1(x)=1

h2(x)=sin(x)

H1(x)=x+C

H2(x)=-cos(x)+C

F(x)=H1(x)+H2(x)

F(x)=x-cos(x)+C