Question

高数积分，求具体过程。

Answer 1

∫(π/4->π/2) [ (sinx)^3- (cosx)^3] sinx cosx dx
-∫(π/2->3π/4) [ (sinx)^3- (cosx)^3] sinx cosx dx

let
y=π-x
dy =-dx

∫(π/2->3π/4) [ (sinx)^3- (cosx)^3] sinx cosx dx
=∫(π/2->π/4) [ (siny)^3+ (cosy)^3] siny cosy dy
=-∫(π/4->π/2) [ (siny)^3 + (cosy)^3] siny cosy dy

∫(π/4->π/2) [ (sinx)^3- (cosx)^3] sinx cosx dx
-∫(π/2->3π/4) [ (sinx)^3- (cosx)^3] sinx cosx dx
=∫(π/4->π/2) [ (sinx)^3- (cosx)^3] sinx cosx dx
+∫(π/4->π/2) [ (siny)^3 + (cosy)^3] siny cosy dy
=2∫(π/4->π/2) (sinx)^3. sinx cosx dx
=2∫(π/4->π/2) (sinx)^4. dsinx
=(2/5)[ (sinx)^5]| (π/4->π/2)
=(2/5) [ 1- 1/2^(5/2) ]