Answer to Question #248621 in Calculus for John

Question

∫ csc2
(2 − 3

Answer 1

$\displaystyle \int\cosec^2xdx\\ \cosec^2x= \frac{\sec^2x}{\tan^2x}\\ \implies \int \frac{\sec^2x}{\tan^2x}dx,\qquad \text{Let u = tanx}\\ \therefore du = \sec^2xdx \implies dx = \frac{du}{sec^2x}\\ \text{Substituting the values above, we have}\\ \int \frac{1}{u^2}du = -u^{-1}+c, \text{ Recall that u = tanx}\\ \therefore \int cosec^2xdx = -cotx + c$