三 2. 微積分基本定理
2.1 令x0 = 0 < x1 = Nπ < x2 = 2πN <· · · < N πN = π.
因為sin x是連續函數, 所以其黎曼和會收斂, 因此可取ξi = xi去計算其收斂值,
∴∑N
i=1f (ξi)M xi =∑N
i=1sin(xi)· Nπ =∑N i=1
cos(xi−Mx2 )−cos(xi+Mx2 ) 2 sinMx2
π N
= Nπ 2 sin1 π 2N
∑N
i=1(cos(xi− Mx2 )− cos(xi+Mx2 ))
= Nπ 2 sin1 π 2N
(
cos(x1− Mx2 )− cos(x1 +Mx2 ) + cos(x2− Mx2 )− cos(x2 +Mx2 ) +· · · + cos(xN − Mx2 )− cos(xN − Mx2 )
)
= Nπ 2 sin1 π 2N
(
cos(x1− Mx2 )− cos(xN −Mx2 ) )
(∵ cos(xi−1−Mx2 ) = cos(xi− Mx2 ) )
=
π 2N
sin2Nπ
(
cos(2Nπ )− cos((2N +1)π2N ) )
(N → ∞) = 1 × (cos 0 − cos π) = (1 − (−1)) = 2.
所以∫π
0 sin xdx = 2.
2.2 令F (x) =∫x
a f (x)dx ⇒ F0(x) = f (x)
∫g(x)
h(x) f (x)dx =∫g(x)
a f (x)dx−∫h(x)
a f (x)dx = F (g(x))− F (h(x))
∴ dxd( ∫g(x)
h(x) f (x)dx )
= (
F (g(x))− F (h(x)))0
= (
F (g(x)) )0
−(
F (h(x)) )0
= F0(g(x))g0(x)− F0(h(x))h0(x) = f (g(x))g0(x)− f(h(x))h0(x).
2.3 (1) dxd ( ∫x2
x2 2
ln√ xdx
)
= (ln√
x2)·2x −(ln√
x2
2)·x = x(ln x2−ln√x2) = x ln(√ 2x).
(2) dxd( ∫sin x 1 3x2dx
)
= 3(sin x)2(sin x)0 − 3 · (1)2· (1)0 = 3 sin2x cos x.
(3) dxd( ∫tan x 0
1 1+x2dx
)
= 1+(tan x)1 2(tan x)0 = 1+tan1 2xsec2x = secsec22xx = 1.
(4) dxd( ∫tan−1x
0 sec2xdx
)
= (sec2(tan−1x))· (tan−1x)0 = (sec2(tan−1x))1+x1 2
= (√
1 + x2)2· 1+x12 = 1.
(5) dxd( ∫√x
−√
xsin(x2)dx )
= sin(√
x)2· (√
x)0− sin(−√
x)2· (−√ x)0
= sin x·2√1x − sin x · (−2√1x) = sin x· 2√1x + sin x· 2√1x = √1 xsin x.
(6) dxd( ∫√x
−√
xsin(x3)dx )
= sin(√
x)3· (√
x)0− sin(−√
x)3· (−√ x)0
= sin x3/2· 2√1x − sin(−x3/2)· (−2√1x) = sin(x3/2)· 2√1x− sin(x3/2)· (2√1x) = 0.
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