Calculus Final Exam

Integral of ∫ k dx

k·x + C (where k is a constant)

Integral of ∫ xⁿ dx (n ≠ -1)

(xⁿ⁺¹)/(n+1) + C

Integral of ∫ (f(x) ± g(x)) dx

∫ f(x) dx ± ∫ g(x) dx

∫ k·f(x) dx

c·∫ f(x) dx

∫ eˣ dx

eˣ + C

∫ aˣ dx

aˣ / ln(a) + C (a > 0, a ≠ 1)

∫ 1/x dx

ln|x| + C

∫ sin(x) dx

-cos(x) + C

∫ cos(x) dx

sin(x) + C

∫ sec²(x) dx

tan(x) + C

∫ csc²(x) dx

-cot(x) + C

∫ sec(x)tan(x) dx

sec(x) + C

∫ csc(x)cot(x) dx

-csc(x) + C

∫ f′(x)/f(x) dx

ln|f(x)| + C

What is integration by parts?

∫ u dv = uv - ∫ v du

OR

∫ f’(x) g(x) dx = f(x)g(x) - ∫ f(x)g’(x)

What is u-substitution?

A method where you let u = g(x), and rewrite ∫ f(g(x))g′(x) dx as ∫ f(u) du.

Formula for the average value of a continuous function f(x) on [a, b]

basic formula for the areas between two curves

where f(x) > g(x) on [a,b]

How to find points of intersection for area between curves?

Set f(x)=g(x)f(x) = g(x)f(x)=g(x), solve for x to find the limits of integration.

What is the Fundamental Theorem of Calculus, Part 1

What is the Fundamental Theorem of Calculus, Part 2?