Question

四道高数题急求！！！

Answer 1

Answer 2

(1)
∫ e^x .sinx dx
=∫ sinx de^x
=sinx.e^x -∫ cosx .e^x dx
=sinx.e^x -∫ cosx de^x
=sinx.e^x -cosx. e^x -∫ e^x .sinx dx
2∫ e^x .sinx dx =sinx.e^x -cosx. e^x
∫ e^x .sinx dx =(1/2)[sinx.e^x -cosx. e^x ] +C
(2)
∫ e^x .cosx dx
=∫ cosx de^x
=cosx. e^x + ∫ sinx .e^x dx
=cosx. e^x + ∫ sinx de^x
=cosx. e^x + sinx. e^x -∫ e^x .cosx dx
2∫ e^x .cosx dx =cosx. e^x + sinx. e^x
∫ e^x .cosx dx =(1/2)[cosx. e^x + sinx. e^x] + C
(3)
∫ xlnx
=(1/2)∫ lnx dx^2
=(1/2)x^2.lnx- (1/2)∫ x dx
=(1/2)x^2.lnx- (1/4)x^2 +C
(4)
∫ xe^x dx
=∫ x de^x
=xe^x -∫ e^x dx
=xe^x - e^x +C

Answer 3

∫e^xsinxdx
=∫sinxde^x
=e^xsinx-∫e^xdsinx
=e^xsinx-∫e^xcosxdx
=e^xsinx-∫cosxde^x
=e^xsinx-e^xcosx+∫e^xdcosx
=e^xsinx-e^xcosx-∫e^xsinxdx
所以
∫e^xsinxdx=(e^xsinx-e^xcosx)/2+C
……………
∫e^xcosxdx
=∫e^xd(sinx)
=e^xsinx-∫sinxe^xdx
=e^xsinx+∫e^xd(cosx)
=e^xsinx+e^xcosx-∫e^xcosxdx
所以
2∫e^xcosxdx
=e^xsinx+e^xcosx+c
∫e^xcosxdx=(e^xsinx+e^xcosx)/2 +C
…………
∫xlnxdx

=1/2∫lnx d(x^2) (分部积分)

=1/2(x^2*lnx-∫x^2 d(lnx)

=1/2(x^2*lnx-∫xdx)

=1/2(x^2*lnx-1/2x^2+C)

=1/2x^2*lnx-1/4x^2+C
……………
∫xe^xdx
=∫xde^x
=x*e^x-∫e^xdx
=x*e^x-e^x+c
=(x-1)*e^x+c

Answer 4

Answer 5