高数微积分基本公式

∫ (1 + lnx)/x dx

= ∫ (1 + lnx) d(lnx)

= ∫ (1 + lnx) d(1 + lnx)

= (1 + lnx)²/2 + C

= (1 + 2lnx + ln²x)/2 + C

= lnx + (1/2)ln²x + C''

或

= ∫ (1 + lnx) d(lnx)

= ∫ d(lnx) + ∫ lnx d(lnx)

= lnx + (1/2)ln²x + C

或

令u = lnx，du = (1/x) dx

∫ (1 + lnx)/x dx = ∫ (1 + u)/x (x du)

= ∫ (1 + u) du

= ∫ du + ∫ u du

= u + u²/2 + C

= lnx + (1/2)ln²x + C

lim(x->0) {∫(0,x)[e^(t^2)-1]dt}/(1-cosx)tanx

=lim(x->0) {∫(0,x)[e^(t^2)-1]dt}/[(1/2)x^3]

=lim(x->0) [e^(x^2)-1]/[(3/2)x^2]

=lim(x->0) (x^2)/[(3/2)x^2]

=2/3