4.2 不定积分基本公式

\int

e^{x}

d

x

=

e^{x}

+

C

.

(6) \int \sin x d x = - \cos x + C .

(7) \int \cos x d x = \sin x + C .

(8) \int \sec^{2} x d x = \tan x + C .

(9) \int \csc^{2} x d x = - \cot x + C .

(10) \int \sec x \tan x d x = \sec x + C .

(11) \int \csc x \cot x d x = - \csc x + C .

(12) \int \frac{1}{\sqrt{1 - x^{2}}} d x = \arcsin x + C .

(13) \int \frac{1}{1 + x^{2}} d x = \arctan x + C .

(14) \int \frac{d x}{\sqrt{a^{2} - x^{2}}} = \arcsin \frac{x}{a} + C .

(15) \int \frac{d x}{a^{2} + x^{2}} = \frac{1}{a} \arctan \frac{x}{a} + C .

(16) \int \frac{d x}{x^{2} - a^{2}} = \frac{1}{2 a} \ln ∣ \frac{x - a}{x + a} ∣ + C .

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

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

(19) \int \sec x d x = \ln ∣ \sec x + \tan x ∣ + C .

(20) \int \csc x d x = - \ln ∣ \csc x + \cot x ∣ + C .

\begin{align*} &(1)\int 0\mathrm{d}x = C. \\ &(2)\int x^\alpha \mathrm{d}x = \frac{1}{\alpha+1}x^{\alpha+1}+C \quad (\alpha\neq-1). \\ &(3)\int \frac{1}{x}\mathrm{d}x = \ln|x|+C. \\ &(4)\int a^x\mathrm{d}x = \frac{a^x}{\ln a}+C \quad (a>0,a\neq 1). \\ &(5)\int \mathrm{e}^x\mathrm{d}x = \mathrm{e}^x+C. \\ &(6)\int \sin x\mathrm{d}x = -\cos x+C. \\ &(7)\int \cos x\mathrm{d}x = \sin x+C. \\ &(8)\int \sec^2 x\mathrm{d}x = \tan x+C. \\ &(9)\int \csc^2 x\mathrm{d}x = -\cot x+C. \\ &(10)\int \sec x\tan x\mathrm{d}x = \sec x+C. \\ &(11)\int \csc x\cot x\mathrm{d}x = -\csc x+C. \\ &(12)\int \frac{1}{\sqrt{1-x^2}}\mathrm{d}x = \arcsin x+C. \\ &(13)\int \frac{1}{1+x^2}\mathrm{d}x = \arctan x+C. \\ &(14)\int \frac{\mathrm{d}x}{\sqrt{a^2-x^2}} = \arcsin\frac{x}{a}+C. \\ &(15)\int \frac{\mathrm{d}x}{a^2+x^2} = \frac1a\arctan\frac{x}{a}+C. \\ &(16)\int \frac{\mathrm{d}x}{x^2-a^2} = \frac{1}{2a}\ln\left|\frac{x-a}{x+a}\right|+C. \\ &(17)\int \frac{\mathrm{d}x}{\sqrt{x^2+a^2}} = \ln(x+\sqrt{x^2+a^2})+C. \\ &(18)\int \frac{\mathrm{d}x}{\sqrt{x^2-a^2}} = \ln\left|x+\sqrt{x^2-a^2}\right|+C. \\ &(19)\int \sec x\mathrm{d}x = \ln|\sec x+\tan x|+C. \\ &(20)\int \csc x\mathrm{d}x = -\ln|\csc x+\cot x|+C. \end{align*}

2

d

x

=

\int

(x^{2} - 1 - 1 + \frac{2}{1 + x^{2}})

d

x

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

(3) \int \frac{1 - \sin x}{1 + \sin x} d x & = \int \frac{(1 - \sin x)^{2}}{\cos^{2} x} d x = \int (\sec^{2} x - 2 \sec x \cdot \tan x + \tan^{2} x) d x

= \tan x - 2 \sec x + \tan x - x + C

= 2 \tan x - 2 \sec x - x + C .

\begin{align*} (1)\int \frac{(x+1)^3}{x^2}\mathrm{d}x &=\int \frac{x^3+3x^2+3x+1}{x^2}\mathrm{d}x =\int\left(x+3+\frac{3}{x}+\frac{1}{x^2}\right)\mathrm{d}x \\ &=\frac12x^2+3x+3\ln|x|-\frac1x+C. \\[4pt] (2)\int \frac{x^4-x^2}{1+x^2}\mathrm{d}x &=\int \frac{(x^4-1)-(1+x^2)+2}{1+x^2}\mathrm{d}x =\int\left(x^2-1-1+\frac{2}{1+x^2}\right)\mathrm{d}x \\ &=\frac13x^3-2x+2\arctan x+C. \\[4pt] (3)\int \frac{1-\sin x}{1+\sin x}\mathrm{d}x &=\int \frac{(1-\sin x)^2}{\cos^2x}\mathrm{d}x =\int\left(\sec^2x-2\sec x\cdot\tan x+\tan^2x\right)\mathrm{d}x \\ &=\tan x-2\sec x+\tan x-x+C \\ &=2\tan x-2\sec x-x+C. \end{align*}