Basic formulas

trigonometric 三角函数的

We call $\sqrt{}$ radical.

Reverse power rule

$$\int x^n dx = \frac{x^{n+1}}{n+1} + C$$

You increase the power by one and then divide by the power + 1

Remember that this rule doesn't apply for $n = -1$

$$\begin{align*} \int \sqrt{x} dx &= \int x^\frac{1}{2} dx \\ \\ & = \frac{x^{\frac{1}{2}+1}}{\frac{1}{2}+1} + C \\ \\ & = \frac{x^{\frac{3}{2}}}{\frac{3}{2}} + C \\ \\ & = \sqrt{x^3} \cdot \frac{2}{3} + C \\ \\ & = \frac{2\sqrt{x^3}}{3} + C \end{align*}$$

Indefinite integrals of $sin(x)$, $cos(x)$, and $e^x$

$$\begin{align*} \int \sin(x) dx &= -\cos(x) + C \\ \\ \int \cos(x) dx &= \sin(x) + C \\ \\ \int e^x dx &= e^x + C \end{align*}$$

$$\begin{align*} \\ \\ &\int (\sin{t} + \cos{t} - e^t)dx \\ \\ &= -\cos{t} + \sin{t} - e^t \end{align*}$$

Indefinite integral of $\frac{1}{x}$

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

More trigonometric functions

$$\begin{align*} & \int \sec^2(x) dx = tan(x) + C \\ \\ & \int \sec(x) \tan(x) dx = \sec(x) + C \\ \\ & \int \csc^2(x) dx = -\cot(x) + C \\ \\ & \int \csc(x) \cot(x) dx = -\csc(x) + C \end{align*}$$

Exponential functions

$$\int a^x dx = \frac{a^x}{\ln(a)} + C$$