不定积分中试求一下函数的原函数1.1/(4-x2)^(1/2) 2.1/(4+x2) 3.1/(sinx cosx) 4

1、
Let x = 2sint,dx = 2cost dt
∫ 1/√(4 - x²) dx
= ∫ 1/(2cost) * (2cost) dt
= t + C
= arcsin(x/2) + C
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2、
Let x = 2tant,dx = 2sec²t dt
∫ 1/(4 + x²) dx
= ∫ 1/(4sec²t) * (2sec²t) dt
= (1/2)t + C
= (1/2)arctan(x/2) + C
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3、
∫ 1/(sinxcosx) dx
= ∫ 1/[(sinx)/(cosx) * (cos²x] dx
= ∫ sec²x/tanx dx
= ∫ 1/tanx d(tanx)
= ln(tanx) + C
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4、
∫ tanx dx
= ∫ sinx/cosx dx
= - ∫ 1/cosx d(cosx)
= - ln(cosx) + C
= ln(secx) + C