Calculus Flashcards

Question 1

Integral of $\cot(x)$

Accepted Answer

$\ln|\sin(x)| + C$

Question 2

Integral of $\sec(x)$

Accepted Answer

$\ln|\sec(x) + 	an(x)| + C$

Question 3

Integral of $\csc(x)$

Accepted Answer

$-\ln|\csc(x) + \cot(x)| + C$

Question 4

Integration by Parts Formula

Accepted Answer

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

Question 5

Maclaurin Series for $e^x$

Accepted Answer

$\sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \dots$

Question 6

Maclaurin Series for $\sin(x)$

Accepted Answer

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

Question 7

Maclaurin Series for $\cos(x)$

Accepted Answer

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

Question 8

Maclaurin Series for $\frac{1}{1-x}$

Accepted Answer

$\sum_{n=0}^{\infty} x^n = 1 + x + x^2 + \dots$

Question 9

Fundamental Theorem of Calculus (Part 1)

Accepted Answer

$\frac{d}{dx} \int_a^x f(t) dt = f(x)$

Question 10

Fundamental Theorem of Calculus (Part 2)

Accepted Answer

$\int_a^b f(x) dx = F(b) - F(a)$

Question 11

Limit definition of derivative $f'(x)$

Accepted Answer

$\lim_{h 	o 0} \frac{f(x+h) - f(x)}{h}$

Question 12

L'Hopital's Rule

Accepted Answer

If $\lim \frac{f(x)}{g(x)}$ is $\frac{0}{0}$ or $\frac{\infty}{\infty}$, it equals $\lim \frac{f'(x)}{g'(x)}$

Question 13

Euler's Formula

Accepted Answer

$e^{ix} = \cos(x) + i\sin(x)$

Question 14

Derivative of $c$ (constant)

Accepted Answer

$0$

Question 15

Derivative of $x^n$ (Power Rule)

Accepted Answer

$n x^{n-1}$

Question 16

Derivative of $e^x$

Accepted Answer

$e^x$

Question 17

Derivative of $a^x$

Accepted Answer

$a^x \ln(a)$

Question 18

Derivative of $\ln(x)$

Accepted Answer

$\frac{1}{x}$

Question 19

Derivative of $\log_a(x)$

Accepted Answer

$\frac{1}{x \ln(a)}$

Question 20

Derivative of $\sin(x)$

Accepted Answer

$\cos(x)$

Question 21

Derivative of $\cos(x)$

Accepted Answer

$-\sin(x)$

Question 22

Derivative of $	an(x)$

Accepted Answer

$\sec^2(x)$

Question 23

Derivative of $\sec(x)$

Accepted Answer

$\sec(x)	an(x)$

Question 24

Derivative of $\csc(x)$

Accepted Answer

$-\csc(x)\cot(x)$

Question 25

Derivative of $\cot(x)$

Accepted Answer

$-\csc^2(x)$

Calculus

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