Lớp 12 · 3 thẻ kiến thức
$F$ là một nguyên hàm của $f$ trên $K$ nếu $F'(x)$ $=f(x)$ $,\ \forall x\in K$. Khi đó $\displaystyle\int f(x)\,dx$ $=F(x)+C$.
| Hàm cơ bản | Mở rộng với $ax+b$ |
|---|---|
| $\displaystyle\int x^\alpha dx$ $=\dfrac{x^{\alpha+1}}{\alpha+1}$ $+C$ $(\alpha\ne-1)$ | $\displaystyle\int(ax$ $+b)^\alpha dx$ $=\dfrac1a\cdot\dfrac{(ax+b)^{\alpha+1}}{\alpha+1}$ $+C$ |
| $\displaystyle\int\dfrac1x\,dx$ $=\ln|x|+C$ | $\displaystyle\int\dfrac{dx}{ax+b}$ $=\dfrac1a\ln|ax+b|+C$ |
| $\displaystyle\int e^x\,dx$ $=e^x+C$ | $\displaystyle\int e^{ax+b}dx$ $=\dfrac1ae^{ax+b}+C$ |
| $\displaystyle\int a^x\,dx$ $=\dfrac{a^x}{\ln a}+C$ | $\displaystyle\int a^{kx+b}dx$ $=\dfrac{a^{kx+b}}{k\ln a}+C$ |
| $\displaystyle\int\sin x\,dx$ $=-\cos x+C$ | $\displaystyle\int\sin(ax$ $+b)dx$ $=-\dfrac1a\cos(ax+b)+C$ |
| $\displaystyle\int\cos x\,dx$ $=\sin x+C$ | $\displaystyle\int\cos(ax$ $+b)dx$ $=\dfrac1a\sin(ax+b)+C$ |
| $\displaystyle\int\dfrac{dx}{\cos^2x}$ $=\tan x+C$ | $\displaystyle\int\dfrac{dx}{\cos^2(ax+b)}$ $=\dfrac1a\tan(ax+b)+C$ |
| $\displaystyle\int\dfrac{dx}{\sin^2x}$ $=-\cot x+C$ | $\displaystyle\int\dfrac{dx}{\sin^2(ax+b)}$ $=-\dfrac1a\cot(ax+b)+C$ |