考研基本积分公式汇总

一、前言

在本文中，荒原之梦网汇总整理了考研数学中常用的基本积分公式，可以作为大家求解不定积分和定积分题目的一个参考。

二、正文

下文中出现的 $a$ 和 $C$ 都代表常数，$x$ 代表变量。

$$
\int x^{k} \mathrm{d} x = \frac{1}{k + 1} \cdot x^{k+1} + C, (k \neq -1)
$$

Next

$$
\int \frac{1}{x} \mathrm{d} x = \ln |x| + C
$$

Next

$$
\int a^{x} \mathrm{d} x = \frac{1}{\ln a} \cdot a^{x} + C
$$

$$
\int e^{x} \mathrm{d} x = e^{x} + C
$$

Next

$$
\int \cos x \mathrm{d} x = \sin x + C
$$

$$
\int \sin x \mathrm{d} x = – \cos x + C
$$

Next

$$
\int \frac{1}{\sin x} \mathrm{d} x = \int \csc x \mathrm{d} x = \ln |\csc x – \cot x| + C
$$

$$
\int \frac{1}{\cos x} \mathrm{d} x = \int \sec x \mathrm{d} x = \ln |\sec x + \tan x| + C
$$

Next

$$
\int \frac{1}{\sin^{2} x} \mathrm{d} x = \int \csc^{2} \mathrm{d} x = – \cot x + C
$$

$$
\int \frac{1}{\cos^{2} x} \mathrm{d} x = \int \sec^{2} x \mathrm{d} x = \tan x + C
$$

Next

$$
\int \tan x \mathrm{d} x = – \ln |\cos x| + C
$$

$$
\int \cot x \mathrm{d} x = \ln |\sin x| + C
$$

Next

$$
\int \sec x \tan x \mathrm{d} x = \sec x + C
$$

$$
\int \csc x \cot x \mathrm{d} x = – \csc x + C
$$

Next

$$
\int \frac{1}{a^{2} + x^{2}} \mathrm{d} x = \frac{1}{a} \arctan \frac{x}{a} + C
$$

$$
\int \frac{1}{1 + x^{2}} \mathrm{d} x = \arctan x + C
$$

Next

$$
\int \frac{1}{\sqrt{a^{2} – x^{2}}} \mathrm{d} x = \arcsin \frac{x}{a} + C
$$

$$
\int \frac{1}{\sqrt{1 – x^{2}}} \mathrm{d} x = \arcsin x + C
$$

Next

$$
\int \frac{1}{a^{2} – x^{2}} \mathrm{d} x = \frac{1}{2a} \ln \Big| \frac{a+x}{a-x} \Big| + C
$$

$$
\int \frac{1}{1 -x^{2}} \mathrm{d} x = \frac{1}{2} \ln \Big| \frac{1+x}{1-x} \Big| + C
$$

Next

$$
\int \frac{1}{\sqrt{x^{2} \pm a^{2}}} \mathrm{d} x = \ln |x + \sqrt{x^{2} \pm \ln a^{2}}| + C
$$

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