Answer to Question #178352 in Calculus for Phyroe

Question #178352

Integration Procedures (Integration by Parts)


∫csc^(-2)x dx


1
Expert's answer
2021-04-29T16:56:23-0400

"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})"



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