Question

设f(sinx)=x/sin^2 x 求∫f(x)dx 求帮助啊

Answer 1

f(sinx)=x/sin^2 x

f(x)=arcsinx/x^2
∫f(x)dx

=∫arcsinx/x^2dx
=-∫arcsinxd(-1/x)
=-arcsinx/x+ ∫1/xdarcsinx
=-arcsinx/x+ ∫1/x*1/√(1-x^2)*dx
令x=cost换元求第二个积分啊

Answer 2

let
x =siny
∫f(x)dx
=∫f(siny)d(siny)
= ∫ [y/(siny)^2] d(siny)
=-∫ y d[1/(siny)]
= -y/siny + ∫ (1/siny) dy
=-y/siny + ln|cscy-coty| + C
= -arcsin(x)/x + ln| 1/x - √(1-x^2)/x | + C