4.3 三种主要积分法 | 花野猫的数字花园

4.3 三种主要积分法 | 花野猫的数字花园

x

)

d

x

=

\int

f

[

φ

(

x

)

]

d

φ

(

x

)

=

F

[

φ

(

x

)

]

+

C

.

\int f[\varphi(x)]\varphi'(x)\mathrm{d}x=\int f[\varphi(x)]\mathrm{d}\varphi(x)=F[\varphi(x)]+C.

\sqrt{x}

)

d

(

\sqrt{x}

)

.

(4) & \int f (e^{x}) e^{x} d x = \int f (e^{x}) d (e^{x}) .

(5) & \int f (\ln x) \frac{1}{x} d x = \int f (\ln x) d (\ln x) .

(6) & \int f (\sin x) \cos x d x = \int f (\sin x) d (\sin x) .

(7) & \int f (\cos x) \sin x d x = - \int f (\cos x) d (\cos x) .

(8) & \int f (\tan x) \frac{1}{\cos^{2} x} d x = \int f (\tan x) d (\tan x) .

(9) & \int f (\arcsin x) \frac{1}{\sqrt{1 - x^{2}}} d x = \int f (\arcsin x) d (\arcsin x) .

(10) & \int f (\arctan x) \frac{1}{1 + x^{2}} d x = \int f (\arctan x) d (\arctan x) .

\begin{align*} (1)&\int f(ax+b)\mathrm{d}x=\frac1a\int f(ax+b)\mathrm{d}(ax+b).\\ (2)&\int x^m f(ax^{m+1}+b)\mathrm{d}x=\frac1{(m+1)a}\int f(ax^{m+1}+b)\mathrm{d}(ax^{m+1}+b)\quad(m\neq-1).\\ (3)&\int f(\sqrt{x})\frac{\mathrm{d}x}{\sqrt{x}}=2\int f(\sqrt{x})\mathrm{d}(\sqrt{x}).\\ (4)&\int f(e^x)e^x\mathrm{d}x=\int f(e^x)\mathrm{d}(e^x).\\ (5)&\int f(\ln x)\frac1x\mathrm{d}x=\int f(\ln x)\mathrm{d}(\ln x).\\ (6)&\int f(\sin x)\cos x\mathrm{d}x=\int f(\sin x)\mathrm{d}(\sin x).\\ (7)&\int f(\cos x)\sin x\mathrm{d}x=-\int f(\cos x)\mathrm{d}(\cos x).\\ (8)&\int f(\tan x)\frac1{\cos^2 x}\mathrm{d}x=\int f(\tan x)\mathrm{d}(\tan x).\\ (9)&\int f(\arcsin x)\frac1{\sqrt{1-x^2}}\mathrm{d}x=\int f(\arcsin x)\mathrm{d}(\arcsin x).\\ (10)&\int f(\arctan x)\frac1{1+x^2}\mathrm{d}x=\int f(\arctan x)\mathrm{d}(\arctan x). \end{align*}

2

= \sqrt{3 + 2 x - x^{2}} + \arcsin \frac{x - 1}{2} + C .

\begin{align*} (4)\int \frac{2-x}{\sqrt{3+2x-x^2}}\mathrm{d}x &=\int \frac{(1-x)+1}{\sqrt{3+2x-x^2}}\mathrm{d}x \\ &=\frac12\int \frac{(2-2x)}{\sqrt{3+2x-x^2}}\mathrm{d}x +\int \frac{\mathrm{d}x}{\sqrt{3+2x-x^2}} \\ &=\frac12\int \frac{\mathrm{d}(3+2x-x^2)}{\sqrt{3+2x-x^2}} +\int \frac{\mathrm{d}(x-1)}{\sqrt{4-(x-1)^2}} \\ &=\sqrt{3+2x-x^2}+\arcsin\frac{x-1}{2}+C. \end{align*}

\sin t

+

C

= \ln (x + \sqrt{x^{2} + a^{2}}) - \frac{\sqrt{x^{2} + a^{2}}}{x} + C .

\begin{aligned} \int \frac{\sqrt{x^2+a^2}}{x^2}\mathrm{d}x &=\int \frac{a\sec t}{a^2\tan^2 t}\cdot a\sec^2 t\mathrm{d}t =\int \frac{1}{\sin^2 t\cos t}\mathrm{d}t \\ &=\int \frac{\sin^2 t+\cos^2 t}{\sin^2 t\cos t}\mathrm{d}t \\ &=\int \sec t\mathrm{d}t+\int \frac{\cos t}{\sin^2 t}\mathrm{d}t \\ &=\ln|\sec t+\tan t|-\frac{1}{\sin t}+C \\ &=\ln\left(x+\sqrt{x^2+a^2}\right)-\frac{\sqrt{x^2+a^2}}{x}+C. \end{aligned}

= \ln (x + \sqrt{x^{2} + a^{2}}) - \sqrt{1 + {(\frac{a}{x})}^{2}} + C

= \ln (x + \sqrt{x^{2} + a^{2}}) - \frac{\sqrt{x^{2} + a^{2}}}{x} + C .

\begin{aligned} \int \frac{\sqrt{x^2+a^2}}{x^2}\mathrm{d}x &=\int \frac{x^2+a^2}{x^2\sqrt{x^2+a^2}}\mathrm{d}x \\ &=\int \frac{\mathrm{d}x}{\sqrt{x^2+a^2}}+\int \frac{a^2\mathrm{d}x}{x^3\sqrt{1+\left(\frac{a}{x}\right)^2}} \\ &=\ln\left(x+\sqrt{x^2+a^2}\right)-\frac12\int \frac{\mathrm{d}\left[1+\left(\frac{a}{x}\right)^2\right]}{\sqrt{1+\left(\frac{a}{x}\right)^2}} \\ &=\ln\left(x+\sqrt{x^2+a^2}\right)-\sqrt{1+\left(\frac{a}{x}\right)^2}+C \\ &=\ln\left(x+\sqrt{x^2+a^2}\right)-\frac{\sqrt{x^2+a^2}}{x}+C. \end{aligned}

2

\cos

x

+

2

x

\sin

x

+

2

\cos

x

+

C

.

(3) \int x \ln x d x & = \frac{1}{2} \int \ln x d x^{2} = \frac{1}{2} x^{2} \ln x - \int \frac{1}{2} x^{2} d \ln x = \frac{1}{2} x^{2} \ln x - \frac{1}{2} \int x d x

= \frac{1}{2} x^{2} \ln x - \frac{1}{4} x^{2} + C .

(4) \int e^{x} \sin^{2} x d x & = \int e^{x} \cdot \frac{1}{2} (1 - \cos 2 x) d x = \frac{1}{2} e^{x} - \frac{1}{2} \int e^{x} \cos 2 x d x,

而 \int e^{x} \cos 2 x d x = \frac{1}{5} e^{x} \cos 2 x + \frac{2}{5} e^{x} \sin 2 x + C,

故 \int e^{x} \sin^{2} x d x = \frac{1}{2} e^{x} - \frac{1}{10} e^{x} \cos 2 x - \frac{1}{5} e^{x} \sin 2 x + C .

\begin{align*} (1)\int x\mathrm e^{2x}\mathrm dx &=\frac12\int x\mathrm d\mathrm e^{2x} =\frac12 x\mathrm e^{2x}-\frac12\int \mathrm e^{2x}\mathrm dx =\frac12 x\mathrm e^{2x}-\frac14\mathrm e^{2x}+C.\\[4pt] (2)\int x^2\sin x\mathrm dx &=-\int x^2\mathrm d\cos x =-x^2\cos x+2\int x\cos x\mathrm dx =-x^2\cos x+2\int x\mathrm d\sin x\\ &=-x^2\cos x+2x\sin x+2\cos x+C.\\[4pt] (3)\int x\ln x\mathrm dx &=\frac12\int \ln x\mathrm d x^2 =\frac12 x^2\ln x-\int \frac12 x^2\mathrm d\ln x =\frac12 x^2\ln x-\frac12\int x\mathrm dx\\ &=\frac12 x^2\ln x-\frac14 x^2+C.\\[4pt] (4)\int \mathrm e^x \sin^2 x\mathrm dx &=\int \mathrm e^x\cdot \frac12(1-\cos 2x)\mathrm dx =\frac12\mathrm e^x-\frac12\int \mathrm e^x\cos 2x\mathrm dx,\\ &\text{而}\int \mathrm e^x\cos 2x\mathrm dx=\frac15\mathrm e^x\cos 2x+\frac25\mathrm e^x\sin 2x+C,\\ &\text{故}\int \mathrm e^x \sin^2 x\mathrm dx =\frac12\mathrm e^x-\frac1{10}\mathrm e^x\cos 2x-\frac15\mathrm e^x\sin 2x+C. \end{align*}