高等数学，求定积分…最好有解答过程！谢谢啦

Question

如题所述

Answer 1

(1)
∫(0->π/2) e^(2x). cosx dx
=∫(0->π/2) e^(2x). dsinx
=[ e^(2x). sinx ]|(0->π/2) -2∫(0->π/2) e^(2x). sinx dx
=e^π +2∫(0->π/2) e^(2x). dcosx
=e^π +2[ e^(2x). cosx ]|(0->π/2) -4∫(0->π/2) e^(2x). cosx dx
=e^π -2 -4∫(0->π/2) e^(2x). cosx dx
5∫(0->π/2) e^(2x). cosx dx = e^π -2
∫(0->π/2) e^(2x). cosx dx = (1/5)(e^π -2 )

(2)
∫(0->x) t^2. cos(2t) dt
=(1/2)∫(0->x) t^2. dsin(2t)
=(1/2)[ t^2. sin(2t) ]|(0->x) -∫(0->x) t. sin(2t) dt
=(1/2)x^2. sin(2x) +(1/2)∫(0->x) t. dcos(2t)
=(1/2)x^2. sin(2x) +(1/2)[ t. cos(2t) ]|(0->x) -(1/2)∫(0->x) cos(2t) dt
=(1/2)x^2. sin(2x) +(1/2)xcos(2x) -(1/4) [sin(2t)]|(0->x)
=(1/2)x^2. sin(2x) +(1/2)xcos(2x) -(1/4)sin(2x)追问

谢谢啦