一些简单的练习
常用公式，供反复练习
区分两个函数
指数函数：x在指数上 y = a x y=a^x y=ax
幂函数： y = x a y=x^a y=xa
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基本积分表
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∫ s i n x d x = \int sinx dx= ∫sinxdx=
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∫ 1 x 2 d x = ∫ x − 2 d x = \int\frac{1}{x^2}dx=\int x^{-2}dx= ∫x21dx=∫x−2dx=
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∫ 1 x d x = ∫ x − 1 d x = \int\frac{1}{x}dx=\int x^{-1}dx= ∫x1dx=∫x−1dx=
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∫ 1 x d x = ∫ x − 1 2 d x = \int\frac{1}{\sqrt{x}}dx=\int x^{-\frac{1}{2}}dx= ∫x1dx=∫x−21dx=
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∫ 1 d x = ∫ x 0 d x = \int 1dx=\int x^0dx= ∫1dx=∫x0dx=
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∫ x d x = \int xdx= ∫xdx=
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∫ 2 x d x = \int 2x dx= ∫2xdx=
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∫ x 2 d x = \int x^2dx= ∫x2dx=
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∫ x 3 d x = \int x^3dx= ∫x3dx=
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(幂函数的原函数) ∫ x u d x = \int x^u dx= ∫xudx=
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∫ x 2 x d x = \int x^2\sqrt{x}dx= ∫x2xdx= （幂函数，不妨先写成x的几次方的形式）
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∫ x + 1 d x = \int x+1 dx= ∫x+1dx=
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∫ 2 x + 1 d x = \int 2x+1 dx= ∫2x+1dx=
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∫ s e c 2 x d x = \int sec^2xdx= ∫sec2xdx=
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∫ 1 c o s 2 x d x = \int \frac{1}{cos^2x}dx= ∫cos2x1dx=
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∫ d x 1 + x 2 d x = \int \frac{dx}{1+x^2}dx= ∫1+x2dxdx=
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∫ d x 1 − x 2 = \int \frac{dx}{\sqrt{1-x^2}}= ∫1−x2dx=
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∫ s i n 2 x d x = \int sin^2xdx= ∫sin2xdx=
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∫ 1 s i n 2 x d x = \int \frac{1}{sin^2x}dx= ∫sin2x1dx=
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∫ s e c x t a n x d x = \int secxtanxdx= ∫secxtanxdx=
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∫ c s c x c o t x d x = \int cscxcotxdx= ∫cscxcotxdx=
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∫ e x d x = \int e^xdx= ∫exdx=
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∫ a x d x = \int a^xdx= ∫axdx=
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∫ t a n 2 x d x = \int tan^2xdx= ∫tan2xdx=
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∫ t a n x d x = \int tanxdx= ∫tanxdx=
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∫ c o t x d x = \int cotxdx= ∫cotxdx=
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∫ s e c x d x = \int secxdx= ∫secxdx=
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∫ c s c x d x = \int cscxdx= ∫cscxdx=
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∫ d x a 2 + x 2 d x = \int \frac{dx}{a^2+x^2}dx= ∫a2+x2dxdx=
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∫ d x x 2 − a 2 d x = \int \frac{dx}{x^2-a^2}dx= ∫x2−a2dxdx=
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∫ d x a 2 − x 2 = \int \frac{dx}{\sqrt{a^2-x^2}}= ∫a2−x2dx=
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∫ d x x 2 + a 2 = \int \frac{dx}{\sqrt{x^2+a^2}}= ∫x2+a2dx=
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∫ d x x 2 − a 2 = \int \frac{dx}{\sqrt{x^2-a^2}}= ∫x2−a2dx=
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∫ k f ( x ) d x = k ∫ f ( x ) d x \int kf(x)dx=k\int f(x)dx ∫kf(x)dx=k∫f(x)dx
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最后几题答案：
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换元法：
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∫ 2 c o s 2 x d x = \int 2cos2xdx= ∫2cos2xdx=
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∫ 1 3 + 2 x d x = \int \frac{1}{3+2x}dx= ∫3+2x1dx=
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∫ x 2 ( x + 2 ) 3 = \int \frac{x^2}{(x+2)^3}= ∫(x+2)3x2=
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∫ 2 x e x 2 = \int 2xe^{x^2}= ∫2xex2=
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∫ x 1 − x 2 d x = \int x\sqrt{1-x^2}dx= ∫x1−x2dx=
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三角函数二倍角公式
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c o s 2 x = 1 − 2 s i n 2 x = 2 c o s 2 x − 1 = c o s 2 x − s i n 2 x cos2x=1-2sin^2x=2cos^2x-1=cos^2x-sin^2x cos2x=1−2sin2x=2cos2x−1=cos2x−sin2x
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s i n 2 x = 2 s i n x c o s x sin2x=2sinxcosx sin2x=2sinxcosx
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二倍角公式的练习
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∫ s i n 3 x d x = \int sin^3xdx= ∫sin3xdx=
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∫ c o s 2 x 2 d x = \int cos^2\frac{x}{2}dx= ∫cos22xdx=
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求微分练习
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d x 2 = dx^2= dx2=
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d x 3 = dx^3= dx3=
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d 1 x = d\frac{1}{x}= dx1=
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d ( 3 x + 5 ) = d(3x+5)= d(3x+5)=
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d ( 3 x ) = d(3x)= d(3x)=
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d ( 6 x ) = d(6x)= d(6x)=
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d 1 x 2 = d\frac{1}{x^2}= dx21=
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d l n x = dlnx= dlnx=
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d e x = de^x= dex=
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d c o s x = dcosx= dcosx=
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d s i n x = dsinx= dsinx=
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d s e c x = dsecx= dsecx=
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d c s c x = dcscx= dcscx=
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d t a n x = dtanx= dtanx=
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d a r c t a n x = darctanx= darctanx=
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d a r c s i n x = darcsinx= darcsinx=
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d a r c c o s x = darccosx= darccosx=
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d x = d\sqrt{x}= dx=
