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Integral rules

27 formulas · what each one means · a free lesson for each

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Basic integrals

Power rule
∫xn dx = xn+1n + 1 + C, n ≠ −1
Add 1 to the power, divide by the new power.Lesson: Antiderivatives →
1/x
∫1x dx = ln|x| + C
Lesson: Integrals with e and ln →
eˣ
∫ex dx = ex + C ∫ekx dx = 1kekx + C
Lesson: Integrals with e and ln →
sin and cos
∫sin x dx = −cos x + C ∫cos x dx = sin x + C
Lesson: Antiderivatives →
sec²
∫sec2 x dx = tan x + C
Lesson: Antiderivatives →
Constant multiple and sum
∫(af + bg) dx = a∫f dx + b∫g dx
Lesson: Antiderivatives →

Definite integrals

Fundamental theorem
∫ba f(x) dx = F(b) − F(a)
F is an antiderivative of f.Lesson: Definite integrals and area →
Fundamental theorem, part 1
ddx∫xa f(t) dt = f(x)
Lesson: The Fundamental Theorem, part 1 →
Riemann sum
Σ f(xi) Δx, Δx = b − an
Rectangles under the curve.Lesson: Riemann sums →
Area between curves
∫ba (top − bottom) dx
Lesson: Area between two curves →
Average value
1b − a∫ba f(x) dx
Lesson: Average value of a function →
Net change
∫ba F′(x) dx = F(b) − F(a)
Total change from the rate.Lesson: Net change →
Initial value problem
y = ∫f(x) dx, then find C from y(a)
Lesson: Initial value problems →

Integration techniques

u-substitution
∫f(g(x))g′(x) dx = ∫f(u) du, u = g(x)
Lesson: u-substitution →
Integration by parts
∫u dv = uv − ∫v du
Lesson: Integration by parts →
Partial fractions
px + q(x − a)(x − b) = Ax − a + Bx − b
Lesson: Partial fractions →
Trig substitution
√a2 − x2: x = a sin θ √a2 + x2: x = a tan θ √x2 − a2: x = a sec θ
Lesson: Trig substitution →
Improper integral
∫∞a f(x) dx = limb → ∞∫ba f(x) dx
Lesson: Improper integrals →

Power-reducing (half-angle) identities

sin²
sin2 x = 1 − cos 2x2
Used to integrate even powers of sin.Lesson: Trig integrals →
cos²
cos2 x = 1 + cos 2x2
Used to integrate even powers of cos.Lesson: Trig integrals →

Volume, length and area

Disk
V = π∫ba r2 dx
Lesson: Volumes by disks and washers →
Washer
V = π∫ba (R2 − r2) dx
R = outside radius, r = inside radius.Lesson: Volumes by disks and washers →
Shell
V = 2π∫ba r h dx
r = distance from the axis, h = height of the shell.Lesson: Volumes by shells →
Arc length
L = ∫ba √1 + (f′(x))2 dx
Lesson: Arc length →
Surface area of revolution
S = 2π∫ba f(x)√1 + (f′(x))2 dx
Lesson: Surface area of revolution →
Polar area
A = 12∫βα r2 dθ
Lesson: Area in polar coordinates →
Parametric slope
dydx = dy/dtdx/dt
Lesson: Parametric curves →

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