integrate (1-cosx)csc^2 xdx

Question

integrating trigonometric functions

Answer 1

∫(1-cos(x))csc²(x)dx=

∫(csc²(x)-cot(x)csc(x))dx=

-cot(x)+csc(x)+C where C is constant of integration.

[Let u=cot(x)=1/tan(x),

du/dx=-(1/tan²(x))sec²(x)=-(cos²(x)/sin²(x))(1/cos²(x))=-1/sin²(x)=-csc²(x).

So integral of -csc²(x)=cot(x).

Let u=csc(x)=1/sin(x),

du/dx=-(1/sin²(x))cos(x)=-(cos(x)/sin(x))(1/sin(x))=-cot(x)csc(x).

So integral of -cot(x)csc(x)=csc(x).]