微积分2辅导

Calc 2 Tutor

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Calc 2 Tutor

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Integration Rules

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

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

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

$\int \sin(x) dx = -\cos(x) + C$

$\int \cos(x) dx = \sin(x) + C$

By Parts:

$\int u \, dv = uv - \int v \, du$

Series Tests

Ratio Test:

$L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$

If $L < 1$ converges, $L > 1$ diverges.

p-Series:

$\sum \frac{1}{n^p}$ converges if $p > 1$.

Taylor Series

$e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}$

$\sin(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}$

$\cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}$

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