3.∫cosx/(sinx+cosx) dx
= (1/2)∫[(cosx+sinx)+(cosx-sinx)]/(sinx+cos)] dx
= (1/2)∫ dx + (1/2)∫(cosx-sinx)/(sinx+cosx) dx
= x/2 + (1/2)∫d(sinx+cosx)/(sinx+cosx)
= (1/2)(x+ln|sinx+cosx|) + C
4.
令√x=t 则x=t²
dx=2tdt
∫e^√x dx
=∫2t *e^t dt
=∫2t d(e^t)
分部积分
=2te^t-2∫e^tdt
=2te^t-2e^t+C
=2(t-1)e^t+C
t=√x代回
∫e^√xdx=2(√x -1)e^√x+C
代入上下限(4,0)
=2(2-1)e²-2(0-1)e^0
=2e²+2
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