考研数学笔记 – 积分公式
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$$ \int x^k \, dx = \dfrac{1}{k + 1} x^{k + 1} + C $$

$$ \int \dfrac{1}{x} \, dx = \ln \left| x \right| + C $$

$$ \int a^x \, dx = \dfrac{1}{\ln a} a^x + C \; \left( a > 0 且 a \neq 1 \right) $$

$$ \int e^x \, dx = e^x + C $$


$$ \int \sin x \, dx = -\cos x + C $$

$$ \int \cos x \, dx = \sin x + C $$

$$ \int \tan x \, dx = -\ln \left| \cos x \right| + C $$

$$ \int \cot x \, dx = \ln \left| \sin x \right| + C $$

$$ \int \sec x \, dx = \ln \left| \sec x + \tan x \right| + C $$

$$ \int \csc x \, dx = \ln \left| \csc x - \cot x \right| + C $$


$$ \int \sec^2 x \, dx = \tan x + C $$

$$ \int \csc^2 x \, dx = -\cot x + C $$

$$ \int \sec x \tan x \, dx = \int \dfrac{\sin x}{\cos^2 x} \, dx = \sec x + C $$

$$ \int \csc x \cot x \, dx = \int \dfrac{\cos x}{\sin^2 x} = -\csc x + C $$


$$ \int \dfrac{1}{\sqrt{1 - x^2}} \, dx = \arcsin x + C $$

$$ \int \dfrac{1}{\sqrt{a^2 - x^2}} \, dx = \arcsin \dfrac{x}{a} + C $$

$$ \int \dfrac{1}{\sqrt{x^2 + a^2}} \, dx = \ln \left| x + \sqrt{x^2 + a^2} \right| + C $$

$$ \int \dfrac{1}{\sqrt{x^2 - a^2}} \, dx = \ln \left| x + \sqrt{x^2 - a^2} \right| + C $$


$$ \int \dfrac{1}{1 + x^2} \, dx = \arctan x + C $$

$$ \int \dfrac{1}{a^2 + x^2} \, dx = \dfrac{1}{a} \arctan \dfrac{x}{a} + C $$


$$ \int \dfrac{1}{a^2 - x^2} \, dx = \dfrac{1}{2a} \ln \left| \dfrac{a + x}{a - x} \right| + C $$

$$ \int \dfrac{1}{x^2 - a^2} \, dx = \dfrac{1}{2a} \ln \left| \dfrac{x - a}{x + a} \right| + C $$


$$ \int \sqrt{a^2 - x^2} \, dx = \dfrac{a^2}{2} \arcsin \dfrac{x}{a} + \dfrac{x}{2} \sqrt{a^2 - x^2} + C $$


$$
\begin{cases}
I_n = \int_0^{\frac{\pi}{2}} \sin^n x \, dx = \int_n^{\frac{\pi}{2}} \cos^n x \, dx \
\
I_n = \frac{n - 1}{n} I_{n - 2} \
\
I_0 = \dfrac{\pi}{2} \; , \; I_1 = 1
\end{cases}
$$

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