Calc 1 Unit 4

\int_{}^{}0\!\,dx

c

\int Kdx

Kx+c

\int x^{n}dx

\frac{1}{n+1}x_{}^{n+1}+c

\int Kf\left(x\right)dx

$K\int f\left(x\right)\differentialD x

\int f\left(x\right)\pm g\left(x\right)dx

\int f\left(x\right)\differentialD x\pm\int g\left(x\right)\differentialD x

\int\cos\left(x\right)

\sin x+c

\int\sin\left(x\right)dx

-\cos x+c

\int\sec^2\left(x\right)dx

\tan x+c

\int\csc\left(x\right)\tan\left(x\right)dx

-\csc x+c

\int\sec\left(x\right)\tan\left(x\right)dx

\sec x+c

\int\csc^2\left(x\right)dx

-\cot x+c

\int f\left(g\left(x\right)\right)g^{\prime}\left(x\right)dx

F\left(g\left(x\right)\right)+c

\sum_{i=1}^{n}K\cdot a_{I}

K\sum_{i=1}^{n}a_{i}

\sum_{i=1}^{n}a_{i}\pm b_{i}

\sum_{i=1}^{n}a_{i}\pm\sum_{i=1}^{n}b_{i}

\sum_{i=1}^{n}c

c\cdot n

\sum_{i=1}^{n}i

\frac{n\left(n+1\right)}{2}

\sum_{i=1}^{n}i^2

\frac{n\left(n+1\right)\left(2n+1\right)}{6}

\sum_{i=1}^{n}i^3

\frac{n^2\left(n+1\right)^2}{4}

\Delta x=

\frac{b-a}{n}

c_{i}=

a+\Delta xi

f\left(c_{i}\right)=

plug in c_i

Area limit definition

\lim_{n\to\infty}\sum_{i=1}^{n}f\left(c_{i}\right)\Delta x

Define Definite Integrals

\lim_{\left\Vert\Delta\right\Vert\to0}\sum_{i=1}^{n}f\left(c_{i}\right)\Delta x=\int_{a}^{b}\!f\left(x\right)\,dx

Define the fundamental theorem of calculus

\int_{a}^{b}\!f\left(x\right)\,dx=F\left(b\right)-F\left(a\right)

Write equation for u substitution in definite integrals

\int_{a}^{b}\!f\left(g\left(x\right)\right)g^{\prime}\left(x\right)\,dx=\int_{g\left(a\right)}^{g\left(b\right)}\!f\left(u\right)\,du

\int_{a}^{a}\!f\left(x\right)\,dx

0

\int_{a}^{b}\!f\left(x\right)\,dx

-\int_{b}^{a}\!f\left(x\right)\,dx

\int_{a}^{b}\!f\left(x\right)\,dx when c is in (a,b)

\int_{a}^{c}\!f\left(x\right)\,dx+\int_{c}^{b}\!f\left(x\right)\,dx