AP Calculus BC Study Guide with
Complete Solutions
<a,b> ° <c,d> = - ANSWER-ac + bd
∫1 / (√(a² - x²)) dx = - ANSWER-Arcsin (x / a) + C
∫1 / (a² + x²) dx = - ANSWER-(1 / a)(Arctan (x / a)) + C
∫1 / (x * √(x² - a²)) = - ANSWER-(1 / a)(Arcsec (|x| / a)) + C
= (1 / a)(Arccos |a / x|) + C
∫1 / x dx = - ANSWER-ln |x| + C
∫a dx = - ANSWER-ax + C
∫a^x dx = - ANSWER-a^x / ln a + C
∫a→b f(x) dx is an improper integral if... - ANSWER-i) f becomes infinite at one or more
points of the interval of integration, or
ii) one or both of the limits of integration is infinite, or
iii) both (i) and (ii) hold.
∫cos x dx = - ANSWER-sin x + C
∫cot x dx = - ANSWER-ln |sin x| + C
∫csc x dx = - ANSWER--ln |csc x + cot x| + C
∫csc² x dx = - ANSWER--cot x + C
∫cscxcotx dx = - ANSWER--csc x + C
∫e^x dx = - ANSWER-e^x + C
∫ln x dx = - ANSWER-xln x - x + C
∫sec x dx = - ANSWER-ln |sec x + tan x| + C
∫sec² x dx = - ANSWER-tan x + C
∫secxtanxdx = - ANSWER-sec x + C
∫sin x dx = - ANSWER--cos x + C
, ∫tan x dx = - ANSWER-ln |sec x| + C
= -ln |cos x| + C
∫tan² x dx = - ANSWER-tan x - x + C
∫x^n dx = - ANSWER-(x^n+1) / (n + 1) + C,
n≠1
1 + cot² θ = - ANSWER-csc² θ
1 + tan² θ = - ANSWER-sec² θ
A function f(x) is periodic with period p(p > 0) if... - ANSWER-f(x + p) = f(x) for every
value of X.
A function y = f(x) is continuous at x = a if... - ANSWER-i) f(a) exists
ii) lim (x→a) f(x) exists
iii) lim (x→a) f(x) = f(a).
Otherwise, f is discontinuous at x = a.
A function y = f(x) is even if... - ANSWER-f(-x) = f(x) for every X in the function's domain.
Every even function is symmetric about the y-axis.
A function y = f(x) is odd if... - ANSWER-f(-x) = -f(x) for every X in the function's domain.
Every odd function is symmetric about the origin.
A line x = a is a vertical asymptote of the graph y = f(x) if either... - ANSWER-lim(x→a⁺)
f(x) = ±∞
or lim(x→a⁻) f(x) = ±∞.
(Values that make the denominator 0 but not the numerator)
A line y = b is a horizontal asymptote of the graph y = f(x) if either... - ANSWER-
lim(x→∞) f(x) = b
or lim(lim(x→-∞) f(x) = b.
(Compare degrees of functions in faction)
Acceleration Vector: - ANSWER-a(t) = <d²x/dt²,d²y/dt²>
All power functions grow (faster/slower) than any exponential function (a^x, a > 1). -
ANSWER-slower
Among exponential functions, those with larger bases grow (faster/slower) than those
with smaller bases. - ANSWER-faster
Complete Solutions
<a,b> ° <c,d> = - ANSWER-ac + bd
∫1 / (√(a² - x²)) dx = - ANSWER-Arcsin (x / a) + C
∫1 / (a² + x²) dx = - ANSWER-(1 / a)(Arctan (x / a)) + C
∫1 / (x * √(x² - a²)) = - ANSWER-(1 / a)(Arcsec (|x| / a)) + C
= (1 / a)(Arccos |a / x|) + C
∫1 / x dx = - ANSWER-ln |x| + C
∫a dx = - ANSWER-ax + C
∫a^x dx = - ANSWER-a^x / ln a + C
∫a→b f(x) dx is an improper integral if... - ANSWER-i) f becomes infinite at one or more
points of the interval of integration, or
ii) one or both of the limits of integration is infinite, or
iii) both (i) and (ii) hold.
∫cos x dx = - ANSWER-sin x + C
∫cot x dx = - ANSWER-ln |sin x| + C
∫csc x dx = - ANSWER--ln |csc x + cot x| + C
∫csc² x dx = - ANSWER--cot x + C
∫cscxcotx dx = - ANSWER--csc x + C
∫e^x dx = - ANSWER-e^x + C
∫ln x dx = - ANSWER-xln x - x + C
∫sec x dx = - ANSWER-ln |sec x + tan x| + C
∫sec² x dx = - ANSWER-tan x + C
∫secxtanxdx = - ANSWER-sec x + C
∫sin x dx = - ANSWER--cos x + C
, ∫tan x dx = - ANSWER-ln |sec x| + C
= -ln |cos x| + C
∫tan² x dx = - ANSWER-tan x - x + C
∫x^n dx = - ANSWER-(x^n+1) / (n + 1) + C,
n≠1
1 + cot² θ = - ANSWER-csc² θ
1 + tan² θ = - ANSWER-sec² θ
A function f(x) is periodic with period p(p > 0) if... - ANSWER-f(x + p) = f(x) for every
value of X.
A function y = f(x) is continuous at x = a if... - ANSWER-i) f(a) exists
ii) lim (x→a) f(x) exists
iii) lim (x→a) f(x) = f(a).
Otherwise, f is discontinuous at x = a.
A function y = f(x) is even if... - ANSWER-f(-x) = f(x) for every X in the function's domain.
Every even function is symmetric about the y-axis.
A function y = f(x) is odd if... - ANSWER-f(-x) = -f(x) for every X in the function's domain.
Every odd function is symmetric about the origin.
A line x = a is a vertical asymptote of the graph y = f(x) if either... - ANSWER-lim(x→a⁺)
f(x) = ±∞
or lim(x→a⁻) f(x) = ±∞.
(Values that make the denominator 0 but not the numerator)
A line y = b is a horizontal asymptote of the graph y = f(x) if either... - ANSWER-
lim(x→∞) f(x) = b
or lim(lim(x→-∞) f(x) = b.
(Compare degrees of functions in faction)
Acceleration Vector: - ANSWER-a(t) = <d²x/dt²,d²y/dt²>
All power functions grow (faster/slower) than any exponential function (a^x, a > 1). -
ANSWER-slower
Among exponential functions, those with larger bases grow (faster/slower) than those
with smaller bases. - ANSWER-faster
