Question

高等数学中一个积分公式怎么用分部积分求

Answer 1

(1)
∫x/(sinx)^2 dx
=∫x(cscx)^2 dx
=-∫xdcotx
=-xcotx+ ∫cotx dx
=-xcotx + ln|sinx| + C

∫(π/4->3π/4) x/(sinx)^2 dx
=[-xcotx + ln|sinx|]|(π/4->3π/4)
=[ 3π/4 -(1/2)ln2 ] -[-π/4 - (1/2)ln2 ]
=π

(2)
∫e^x.cos2x dx
=∫cos2x de^x
=e^x.cos2x +2∫e^x.sin2x dx
=e^x.cos2x +2∫sin2x de^x
=e^x.cos2x +2sin2x.e^x -4∫e^x.cos2x dx
5∫e^x.cos2x dx = e^x.cos2x +2sin2x.e^x
∫e^x.cos2x dx = (1/5) [e^x.cos2x +2sin2x.e^x ] + C

∫(0->π/2) e^x.cos2x dx
=(1/5) [e^x.cos2x +2sin2x.e^x ]| (0->π/2)
=(1/5) [ -e^(π/2) -1 ]
= -(1/5) (e^(π/2) +1 )

(3)
∫x[sin(x/2)]^2 dx
=(1/2)∫x(1-cosx) dx
=(1/4)x^2 -(1/2)∫xcosx dx
=(1/4)x^2 -(1/2)∫xdsinx
=(1/4)x^2 -(1/2)x.sinx +(1/2)∫sinx dx
=(1/4)x^2 -(1/2)x.sinx -(1/2)cosx + C

(4)
∫(x^2+1).e^(-x) dx
=-e^(-x) +∫x^2.e^(-x) dx
=-e^(-x) -∫x^2.de^(-x)
=-e^(-x) -x^2.e^(-x) +2∫x.e^(-x)dx
=-e^(-x) -x^2.e^(-x) -2∫xde^(-x)
=-e^(-x) -x^2.e^(-x) -2x.e^(-x) +∫e^(-x)dx
=-e^(-x) -x^2.e^(-x) -2x.e^(-x) -e^(-x)dx + C
=-x^2.e^(-x) -2x.e^(-x) -2e^(-x)dx + C

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

Answer 2

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