Memorized Formula Sheet PDF

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Math 101 Formula’s to Memorize Note: updated as we progress. Some prerequisite material is not listed but still assumed that you know. For example, we assume you know all the derivatives from Calculus I and the derivative rules along with the limit definition of the derivative. As another example, √ we assume you know how to evaluate expressions like cos (5π/6) = − 3/2 as the exact form. Trigonometric Formulæ Basic Identities • tan2 (θ) + 1 = sec2 (θ ) Double Angle Identities 1 (1 − cos(2θ )) 2 1 • cos2 (θ) = (1 + cos(2θ)) 2

• sin2 (θ) =

Integral Formulæ Definition of integral Z b n X f (x)dx = lim • f (a + k∆x)∆x,

where ∆x =

n→∞

a

k=1

b−a n

Fundamental Theorem of Calculus Z x  d f (t)dt = f (x) • dx a  Z b b ′  • F (x)dx = F (x) = F (b) − F (a) a

a

Linearity Z b Z b Z • (af (x) + bg(x)) dx = a f (x)dx + b a

a

Area between curves Z b (f (x) − g(x))dx, • Area =

b

g(x)dx a

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

a

Substitution Z Z b f (g (x))g ′ (x)dx = • a

g(b)

f (u)du,

where u = g(x)

g(a)

Integration by parts b Z b Z b  • u(x)v ′ (x)dx = u(x)v (x) − v(x)u′ (x)dx a

a

a

Antiderivatives Powers Z xn+1 • xn dx = + C, n+1 Z 1 dx = ln |x| + C • x

n 6= −1

Trigonometric Z • sin(x)dx = − cos(x) + C •

Z

cos(x)dx = sin(x) + C

•

Z

sec2 (x)dx = tan(x) + C

•

Z

csc2 (x)dx = − cot(x) + C

•

Z

sec(x) tan(x)dx = sec(x) + C

•

Z

csc(x) cot(x)dx = − csc(x) + C

•

Z

tan(x)dx = ln | sec(x)| + C

•

Z

cot(x)dx = ln | sin(x)| + C

Exponentials Z • ex dx = ex + C •

Z

ax dx =

ax + C, ln(a)

a > 0, a 6= 1...