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# 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

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