MIDTERM 1

most general antiderivative

F(x) + C

What is the particular antiderivative of cf(x) ?

cF(x)

• constant multiple rule

What is the particular antiderivative of f(x) + g(x) ?

F(x) + G(x)

What is the particular antiderivative of xⁿ (n ≠ 1) ?

(xⁿ⁺¹)/(n + 1)

• power rule

• useful for polynomials

What is the particular antiderivative of 1/x ?

ln |x|

What is the particular antiderivative of e^x ?

e^x

What is the particular antiderivative of e^cx ?

(1/c)e^cx

What is the particular antiderivative of cos x ?

sin x

What is the particular antiderivative of sin x ?

-cos x

What is the particular antiderivative of sec²x ?

tan x

What is the particular antiderivative of sec x tan x ?

sec x

What is the particular antiderivative of (1)/(1 + x²)

tan^-1x

• arctan x

What is the area of a rectangle?

A = ab

What is the area of a triangle?

A = (1/2)bh

What is the area of a circle?

A = πr^(2)

What is this property of definite integrals?

order of integration

• add a negative

• switch a & b when rewriting integral

• line is going backwards

What is this property of definite integrals?

zero width integral

• limits of integration are = → singular point on graph → ≠ area

What is this property of definite integrals?

constant multiple

• move the constant to the front

What is this property of definite integrals?

sum/difference

• split up functions when they are added or subtracted together

What is this property of definite integrals?

additivity

• Atot = A1 + A2 +…

• area of functions can be split into subintervals & added together to get the total area

What is this property of definite integrals?

min-max inequality

• fmin is the lowest point of the function

• fmax is the highest point of the function

What is this property of definite integrals?

domination

• f(x)dx is always greater than g(x)dx