Unformatted text preview: Derivatives
1
x 1
x ln a [xn ] = nxn−1 [ln x] = [ex ] = ex [ax ] = [ax ] ln a [cos x] = − sin x
[csc x] = − csc x cot x [sin x] = cos x
[sec x] = sec x tan x [tan x] = sec2 x
[cot x] = csc2 x 1
[sin−1 x] = √1−x2
1
[csc−1 x] = − x√x2 −1 1
[cos−1 x] = − √1−x2
1
[sec−1 x] = x√x2 −1 1
[tan−1 x] = 1+x2 x
1
[tan−1 x] = − 1+x2 x [cf ] = cf [f g ] = f g + f g chain rule: [f (g (x))] = f (g (x))g (x) [loga x] = f
g f g −f g
g2 = Indeﬁnite Integrals
integration by parts:
un+1
n+1 un du = +C udv = uv −
du
u , n = −1 v du = ln u + C
au
ln a eu du = eu + C au du = sin udu = − cos u + C
tan udu = − ln cos u + C
sec udu = ln(sec u + tan u) + C cos udu = sin u + C
cot udu = ln sin u + C
csc udu = ln(csc u − cot u) + C sec2 udu = tan u + C
tan2 udu = tan u − u + C csc2 udu = − cot u + C
cot2 udu = − cot u − u + C du
u2 +a2 du
u2 −a2 = √ du
a2 −u2
√ du
u2 +a2 1
a tan−1 u
a +C = sin−1 u + C
a√
= ln u + u2 + a2 + C √ du
u u2 −a2
√ du
u u2 +a2 = 1
a sec−1 1
= − a ln u
a +C
√
a+ u2 +a2
u = √ du
u2 −a2 1 ln , a > 0, a = 1 u−a
u+a = ln u + √ du
u a2 −u2 +C 1
2a +C 1
= − a ln √ +C u 2 − a2 + C √
a+ a2 −u2
u +C ...
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 Fall '07
 TextbookAnswers
 Derivative

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