Test 1: Important Concepts

Constant: y = h(x) = k

∫ K dx = kx + C

Power: y = h(x) = (x)^P

∫ x^P dx = (x ^P+1) / (P + 1) + C

Natural Exponential: y = h(x) = e^kx

∫ e^kx dx = (e^kx) / (ln |x| × k) + C

General Exponential: y = h(x) = a^kx

∫ a^kx dx = (a^kx) / (ln |x| × k) + C

Natural Log: y = h(x) = 1/x

∫ 1/x dx = ln |x| + C

Sine: y = h(x) = sin(x)

∫ sin(x) dx = -cos(x) + C

Cosine: y = h(x) = cos(x)

∫ cos(x) dx = sin(x) + C

Secant Squared: y = h(x) = sec²(x)

∫ sec²(x) dx = tan(x) + C

Secant-Tan: y = h(x) = sec(x) × tan(x)

∫ sec(x) × tan(x) dx = sec(x) + C

Cosecant Squared: y = h(x) = csc²(x)

∫ csc²(x) dx = -cot(x) + C

Csc-Cot: y = h(x) = csc(x) × cot(x)

∫ csc(x) × cot(x) dx = -csc(x) + C

Constant Multiple Rule: y = h(x) = k × f(x)

∫ k f(x) dx = k ∫ f(x) dx = k × F(x) + C

Sum Rule: y = h(x) = f(x) + g(x)

∫ F(x) + G(x) + C

Difference Rule: y = h(x) = f(x) - g(x)

∫ F(x) - G(x) + C

Area of a Circle

A = π × r²

1 = x² + y²

Area of a Square

A = x²

Area of a Rectangle

A = b × h

Area of a Triangle

A = ½ (b × h)

Δx

(b - a) / n

Left Approximate

Area formula

L ≈ Δx [ f(a) + f(c) ]

Right Approximate

Area formula

Δx ≈ [ f(c) + f(b) ]

Middle Approximate

Area formula

M ≈ Δx [ f(d) + f(e) ]

Fundamental Theorem of Calculus (FTC)

∫ f(x) dx = F(x) + C, F’(x) = f(x)

_a ∫ ^b f(x) dx = F(b) - F(a)

Properties of Definite Integrals

_a ∫ ^a f(x) dx = 0

_a ∫ ^b k × f(x) dx = k × _a ∫ ^b f(x) dx

_a ∫ ^b [ f(x) ± g(x) ] dx = _a ∫ ^b f(x) dx ± _a ∫ ^b g(x) dx

_a ∫ ^b f(x) dx = _a ∫ ^c f(x) dx + _c ∫ ^b f(x) dx

_a ∫ ^b f(x) dx = - _b ∫ ^a f(x) dx