Answer to Question #178352 in Calculus for Phyroe

Question

Integration Procedures (Integration by Parts)

∫csc^(-2)x dx

Answer 1

$I=\int{csc^{-2}x}dx=\int{sin^2x dx}=-\cos{x}\sin{x}-$

$-\int{(-\cos{x})\cos{x}dx}=$

$=-\cos{x}\sin{x}+\int{\cos^2{x}dx}=$

$=-\cos{x}\sin{x}+\int{(1-\sin^2{x})dx}=$

$=-\cos{x}\sin{x}+x-\int{\sin^2{x}dx}=$

$=-\cos{x}\sin{x}+x-I$

$2I=-\cos{x}\sin{x}+x$

$I=\frac{1}{2}(x-\cos{x}\sin{x})$