CELE Engineering Mathematics — Integral CalculusCheat Sheet

Formula

∫ x^n dx = (x^(n+1))/(n+1) + C

Meaning

n = power; C = constant of integration (n ≠ −1)

Watch Out

MUST add C for indefinite integrals; exponent rule: add 1 to power, divide by new power

When To Use

Integrating any polynomial term with constant power

Formula

∫ (1/x) dx = ln|x| + C

Meaning

Natural logarithm of absolute value of x

Watch Out

Must use absolute value |x|; if x < 0, ln gives domain error without absolute bars

When To Use

When integrand is 1/x (e.g., ∫ dx/(5x) = (1/5)ln|x| + C)

Formula

∫ e^x dx = e^x + C

Meaning

e = Euler's number (~2.718)

Watch Out

For e^(kx): ∫ e^(kx) dx = (1/k)e^(kx) + C; don't forget the 1/k factor

When To Use

Exponential function with base e

Formula

∫ a^x dx = (a^x)/(ln a) + C

Meaning

a = constant base; a > 0, a ≠ 1

Watch Out

Denominator is ln(a), NOT log(a); if a = e, formula reduces to e^x

When To Use

Exponential with base other than e (e.g., 2^x, 10^x)

Formula

∫ sin x dx = −cos x + C

Meaning

Integral of sine is negative cosine

Watch Out

NEGATIVE sign before cosine; ∫ sin(kx) dx = −(1/k)cos(kx) + C

When To Use

Trigonometric sine function

Formula

∫ cos x dx = sin x + C

Meaning

Integral of cosine is sine (POSITIVE)

Watch Out

POSITIVE sign; ∫ cos(kx) dx = (1/k)sin(kx) + C

When To Use

Trigonometric cosine function

Formula

∫ tan x dx = −ln|cos x| + C = ln|sec x| + C

Meaning

Two equivalent forms; sec x = 1/cos x

Watch Out

Either form acceptable; both equivalent via logarithm properties

When To Use

Tangent function

Formula

∫ sec^2 x dx = tan x + C

Meaning

Secant squared integrates to tangent

Watch Out

For ∫ sec^2(kx) dx = (1/k)tan(kx) + C

When To Use

When you see sec²x or 1/cos²x

Formula

∫ csc^2 x dx = −cot x + C

Meaning

Cosecant squared integrates to NEGATIVE cotangent

Watch Out

NEGATIVE sign; cot x = cos x / sin x

When To Use

When you see csc²x or 1/sin²x

Formula

∫ sec x tan x dx = sec x + C

Meaning

Product of secant and tangent

Watch Out

Watch for sec(kx)tan(kx): ∫ sec(kx)tan(kx) dx = (1/k)sec(kx) + C

When To Use

When integrand is sec x tan x (recognize the pair)

Formula

∫ csc x cot x dx = −csc x + C

Meaning

Product of cosecant and cotangent

Watch Out

NEGATIVE sign; ∫ csc(kx)cot(kx) dx = −(1/k)csc(kx) + C

When To Use

When integrand is csc x cot x

Formula

∫ (1/√(1−x²)) dx = arcsin x + C

Meaning

Arcsine (inverse sine); domain: −1 < x < 1

Watch Out

Domain: |x| < 1; for ∫ (1/√(a²−x²)) dx = arcsin(x/a) + C

When To Use

When integrand has form 1/√(1−u²) (u = variable or function of x)

Formula

∫ (1/(1+x²)) dx = arctan x + C

Meaning

Arctangent (inverse tangent); domain: all real x

Watch Out

For ∫ (1/(a²+x²)) dx = (1/a)arctan(x/a) + C; no domain restriction

When To Use

When integrand is 1/(1+u²)

Formula

∫ (1/(x√(x²−1))) dx = arcsec|x| + C

Meaning

Arcsecant; domain: |x| > 1

Watch Out

Absolute value critical; domain: |x| > 1

When To Use

When integrand has form 1/(x√(x²−1))

Term

Antiderivative

Example

Antiderivative of f(x) = 2x is F(x) = x² + C

Definition

A function F(x) whose derivative is f(x); i.e., F'(x) = f(x). All antiderivatives differ by a constant.

Term

Indefinite Integral

Example

∫ 3x² dx = x³ + C

Definition

The family of all antiderivatives of f(x), written ∫ f(x) dx = F(x) + C.

Term

Constant of Integration (C)

Example

∫ cos x dx = sin x + C (C could be 0, 5, −3, etc.)

Definition

Arbitrary constant added to indefinite integrals to represent the family of antiderivatives.

Formula

u-substitution: ∫ f(g(x))·g'(x) dx = ∫ f(u) du (where u = g(x))

Meaning

u = inner function; du = u'(x) dx; substitute, integrate, replace u back

Watch Out

MUST convert du from dx; don't forget back-substitution; check that du matches exactly

When To Use

Chain rule in reverse; when integrand has composite function with its derivative present

Formula

Integration by Parts: ∫ u dv = uv − ∫ v du

Meaning

u, v are chosen parts; dv = integrand factor; v = ∫ dv; du = u'dx

Watch Out

Choose u wisely; wrong choice leads to harder integral; may need parts twice; watch signs

When To Use

Product of functions (e.g., xe^x, x sin x, x ln x); LIATE rule chooses u: Logarithm, Inverse trig, Algebra, Trig, Exponential (highest priority = u)

Formula

Partial Fractions: P(x)/Q(x) = A/(x−a) + B/(x−b) + ... (linear factors) or (Ax+B)/(x²+bx+c) (irreducible quadratic)

Meaning

Decompose rational function into sum of simpler fractions; A, B, ... constants found by matching coefficients or substitution

Watch Out

If deg(P) ≥ deg(Q), do long division first; repeated factors: (x−a)² → A/(x−a) + B/(x−a)²; quadratic factors must be irreducible (discriminant < 0)

When To Use

Integrating rational functions (polynomial/polynomial); degree of P < degree of Q

Term

U-Substitution

Example

∫ (2x)(x²+1)³ dx: let u = x²+1, du = 2x dx → ∫ u³ du = u⁴/4 + C = (x²+1)⁴/4 + C

Definition

Technique replacing u = g(x) and du = g'(x) dx to simplify composite functions.

Term

Integration by Parts

Example

∫ x sin x dx: u = x, dv = sin x dx → uv − ∫ v du = −x cos x + ∫ cos x dx = −x cos x + sin x + C

Definition

Method ∫ u dv = uv − ∫ v du for products; LIATE rule prioritizes choice of u.

Term

Partial Fractions

Example

(5x+3)/[(x+1)(x−2)] = A/(x+1) + B/(x−2); solving: A = −8/3, B = 13/3

Definition

Decomposition of rational function into sum of simpler fractions with linear or irreducible quadratic denominators.

Formula

∫_a^b f(x) dx = F(b) − F(a) (Fundamental Theorem of Calculus, Part 1)

Meaning

a, b = bounds; F(x) = antiderivative of f(x); evaluate F at upper bound minus lower bound

Watch Out

Order matters: UPPER minus LOWER (not reversed); bounds are included as equality

When To Use

Any definite integral; compute antiderivative, plug in bounds, subtract

Formula

∫_a^b f(x) dx = − ∫_b^a f(x) dx (Reversing Bounds)

Meaning

Swapping bounds reverses sign

Watch Out

Negative sign appears when bounds reversed

When To Use

Rearranging integral limits

Formula

∫_a^c f(x) dx = ∫_a^b f(x) dx + ∫_b^c f(x) dx (Additivity of Bounds)

Meaning

Split integral at intermediate point b (a < b < c)

Watch Out

Order of bounds: must go left to right; b must lie strictly between a and c

When To Use

When f(x) has discontinuity or change in formula within [a,c]

Formula

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

Meaning

Definite integral of sum = sum of definite integrals; same for scalar multiples

Watch Out

Applies to + and − but not × or ÷

When To Use

Any polynomial or sum of functions

Formula

∫_a^b k·f(x) dx = k ∫_a^b f(x) dx (Constant Multiple)

Meaning

Constant factor k pulls out of integral

Watch Out

k is constant, not function of x

When To Use

When integrand has constant multiplier

Formula

d/dx [∫_a^x f(t) dt] = f(x) (Fundamental Theorem of Calculus, Part 2)

Meaning

Derivative of integral (with variable upper limit) = integrand at that limit

Watch Out

Must use chain rule if upper limit is g(x): d/dx [∫_a^(g(x)) f(t) dt] = f(g(x))·g'(x)

When To Use

Differentiating integrals with variable bounds

Term

Definite Integral

Example

∫_0^2 3x² dx = [x³]_0^2 = 8 − 0 = 8

Definition

∫_a^b f(x) dx represents the signed area under curve f(x) from x = a to x = b; value is a number (not a function).

Term

Fundamental Theorem of Calculus

Example

Part 1: ∫_1^3 2x dx = [x²]_1^3 = 9 − 1 = 8. Part 2: d/dx ∫_0^x sin t dt = sin x

Definition

Part 1: ∫_a^b f(x) dx = F(b) − F(a) if F'(x) = f(x). Part 2: d/dx ∫_a^x f(t) dt = f(x).

Term

Signed Area

Example

∫_0^π sin x dx = 2 (all above axis), but ∫_0^(2π) sin x dx = 0 (halves cancel)

Definition

Area above x-axis counts positive; area below x-axis counts negative.

Formula

A = ∫_a^b [f(x) − g(x)] dx (Area between curves: x-integration, vertical strips)

Meaning

f(x) = upper curve, g(x) = lower curve; a, b = left, right bounds (x-coordinates of intersection)

Watch Out

MUST determine which is upper/lower (test point or sketch); if roles reverse in interval, split integral; order: ALWAYS upper − lower (result ≥ 0)

When To Use

When curves are functions of x; find intersection points (set f = g), integrate upper minus lower

Formula

A = ∫_c^d [x_right(y) − x_left(y)] dy (Area between curves: y-integration, horizontal strips)

Meaning

x_right(y) = rightmost curve, x_left(y) = leftmost curve; c, d = lower, upper y-bounds

Watch Out

RIGHT minus LEFT (opposite of x-method); useful when dy integration simpler or vertical strips awkward

When To Use

When easier to write x as function of y, or when curves are vertical/near-vertical; find intersection (y-values), integrate right minus left

Term

Area Between Curves

Example

Between y = x and y = x² from x = 0 to 1: A = ∫_0^1 (x − x²) dx = [x²/2 − x³/3]_0^1 = 1/2 − 1/3 = 1/6

Definition

Definite integral of the absolute difference between two functions over an interval; represents the total enclosed area.

Term

Intersection Points

Example

y = x² and y = 2x intersect where x² = 2x → x(x−2) = 0 → x = 0 or x = 2

Definition

Points where two curves meet; found by solving f(x) = g(x) (or x_right = x_left for horizontal strips).

Formula

V = π ∫_a^b [R(x)]² dx (Disk Method: revolve about x-axis)

Meaning

R(x) = radius (distance from x-axis to curve); a, b = x-bounds; disk has area πr² at each x

Watch Out

R² not (∫R)²; radius = y-coordinate; if axis is NOT x-axis, R = distance to that axis (e.g., about y = c, use R = |f(x)−c|)

When To Use

Revolving region bounded by y = f(x) ≥ 0, y = 0, x = a, x = b about the x-axis

Formula

V = π ∫_a^b [R_outer²(x) − R_inner²(x)] dx (Washer Method: revolve about x-axis with hole)

Meaning

R_outer = outer radius (top curve), R_inner = inner radius (bottom curve); both squared

Watch Out

BOTH radii squared; π is common factor; R_inner² SUBTRACTED; check which curve is farther from axis

When To Use

Revolving region between two curves (upper and lower) about the x-axis; upper curve must have larger radius

Formula

V = π ∫_c^d [R(y)]² dy (Disk Method: revolve about y-axis)

Meaning

R(y) = radius (distance from y-axis to curve x = f(y)); c, d = y-bounds

Watch Out

Integrate with respect to y; radius = x-coordinate = f(y); washer version subtracts inner²

When To Use

Revolving region bounded by x = f(y) ≥ 0, x = 0, y = c, y = d about the y-axis

Formula

V = 2π ∫_a^b x·f(x) dx (Shell Method: revolve about y-axis)

Meaning

x = radius of cylindrical shell; f(x) = height; shell volume = 2πrh; integrate along x

Watch Out

Factor 2π is part of formula (not added separately); x·f(x) is the integrand; NOT [x·f(x)]²

When To Use

Alternative to disk/washer for y-axis; especially useful when f(x) is hard to invert or bounds awkward for disk method

Formula

V = 2π ∫_c^d y·x(y) dy (Shell Method: revolve about x-axis)

Meaning

y = radius; x(y) = height; integrate with respect to y

Watch Out

Radius = y (distance from x-axis); height = x(y); less common but valid alternative

When To Use

Alternative for revolving about x-axis when washer method is harder

Term

Disk Method

Example

y = √x from 0 to 4 revolved about x-axis: V = π∫_0^4 (√x)² dx = π∫_0^4 x dx = π[x²/2]_0^4 = 8π

Definition

Volume by stacking circular disks perpendicular to axis of revolution; V = π∫R² da (where da is differential along axis).

Term

Washer Method

Example

Between y = x² and y = 2x about x-axis (0 ≤ x ≤ 2): V = π∫_0^2 [(2x)² − (x²)²] dx

Definition

Volume by stacking washers (disks with holes) when region between two curves; V = π∫(R_outer² − R_inner²) da.

Term

Shell Method

Example

y = x² from 0 to 2 revolved about y-axis: V = 2π∫_0^2 x·x² dx = 2π∫_0^2 x³ dx = 8π

Definition

Volume by summing cylindrical shells; V = 2π∫r·h da (r = radius, h = height, da = differential perpendicular to shell).

Formula

M_y = ∫ x·dA = ∫_a^b x·f(x) dx (First Moment about y-axis)

Meaning

M_y = weighted x-position; dA = infinitesimal area element (f(x) dx for vertical strip)

Watch Out

Integrand is x (not x²); dA = f(x) dx for region under single curve; for washer, dA = [R_outer² − R_inner²] dx

When To Use

Computing x-coordinate of centroid; represents 'rotational tendency' about y-axis

Formula

M_x = ∫ y·dA = ∫_a^b (1/2)[f(x)]² dx (First Moment about x-axis, vertical strip)

Meaning

M_x = weighted y-position; (1/2)f(x) is centroid of vertical strip (half height); multiply by height f(x) to get y-contribution

Watch Out

Factor (1/2)f(x) is CRITICAL: centroid of strip is at y/2, so y_centroid_strip × area_strip = (f/2) × f dx = (1/2)f² dx; NOT just ∫ f(x) dx

When To Use

Computing y-coordinate of centroid; vertical strip from y = 0 to y = f(x)

Formula

A = ∫_a^b f(x) dx (Area of region under curve)

Meaning

Total area from x = a to x = b under y = f(x)

Watch Out

Simple definite integral; no x or y weighting

When To Use

Denominator for centroid formulas; also for normalization

Formula

bar(x) = M_y / A = [∫_a^b x·f(x) dx] / [∫_a^b f(x) dx] (x-coordinate of centroid)

Meaning

x-bar = weighted average x-position over the region

Watch Out

Numerator: x·f(x); denominator: f(x) only; don't confuse with moment formula

When To Use

Finding horizontal position of geometric center

Formula

bar(y) = M_x / A = [∫_a^b (1/2)[f(x)]² dx] / [∫_a^b f(x) dx] (y-coordinate of centroid)

Meaning

y-bar = weighted average y-position; (1/2)f² in numerator accounts for strip's own centroid

Watch Out

Numerator is (1/2)f² (NOT just f²); common mistake: using f(x) instead of (1/2)f²

When To Use

Finding vertical position of geometric center

Formula

I_x = ∫ y² dA = ∫_a^b (1/3)[f(x)]³ dx (Second Moment / Moment of Inertia about x-axis, vertical strip)

Meaning

I_x = measure of resistance to rotation about x-axis; (1/3)f(x) is second moment per unit width for vertical strip

Watch Out

Exponent 3 (not 2); (1/3)f³ comes from ∫_0^f y² dy = f³/3 for strip centroid; critical for engineering applications

When To Use

Structural analysis (beams, bending); (1/3)f³ for vertical strip from y = 0 to y = f(x)

Formula

I_y = ∫ x² dA = ∫_a^b x²·f(x) dx (Second Moment / Moment of Inertia about y-axis)

Meaning

I_y = resistance to rotation about y-axis; integrate x²·dA

Watch Out

Integrand is x²·f(x) (not x·f); different from M_y

When To Use

Structural rotation about vertical axis

Formula

I_G = I_parallel − Ad² (Parallel Axis Theorem for composite shapes)

Meaning

I_G = centroidal moment of inertia; I_parallel = moment about any parallel axis; A = area; d = distance between axes

Watch Out

Formula is I_parallel = I_G + Ad² (rearrange: I_G = I_parallel − Ad²); d is distance between axes; A is total area

When To Use

Finding moment of inertia about centroid given I about another parallel axis (or vice versa)

Value

x̄ = b/3 from vertex (along median)

Symbol

Various; depends on orientation

Quantity

Centroid of triangle (base b, height h)

Value

ȳ = 4r/(3π) ≈ 0.424r from diameter

Symbol

Standard tabulation

Quantity

Centroid of semicircle (radius r, diameter on x-axis)

Value

I_base = (1/3)bh³

Symbol

I

Quantity

Second moment of rectangle (width b, height h) about base

Value

I_G = (1/12)bh³

Symbol

I_G

Quantity

Second moment of rectangle about centroid

Term

Centroid

Example

Centroid of rectangle from (0,0) to (4,2) is (2,1)

Definition

Geometric center of a region; point (x̄, ȳ) where the area is 'balanced'; x̄ = M_y/A, ȳ = M_x/A.

Term

First Moment

Example

M_y = ∫_0^2 x·x² dx = ∫_0^2 x³ dx; centroid x-coordinate = M_y / A

Definition

M = ∫ (coordinate)·dA; M_x about x-axis, M_y about y-axis; first moment divided by area gives centroid coordinate.

Term

Second Moment (Moment of Inertia)

Example

I_x = ∫_0^2 (1/3)(x²)³ dx for curve y = x² from 0 to 2

Definition

I = ∫ (coordinate)² dA; I_x about x-axis, I_y about y-axis; measures resistance to rotation.

Term

Parallel Axis Theorem

Example

Rectangle 2×4: I_G about centroid = (1/12)×2×4³ = 10.67; about corner 2 units away: I = 10.67 + 8×4 = 42.67

Definition

Relates moment of inertia about centroid (I_G) to moment about any parallel axis: I_parallel = I_G + Ad².

Formula

L = ∫_a^b √[1 + (dy/dx)²] dx (Arc Length: curve y = f(x) from x = a to x = b)

Meaning

dy/dx = f'(x); integrand √[1 + (f')²] ds element; L = total distance along curve

Watch Out

Integrand is √[1 + (dy/dx)²], NOT √[(dy/dx)²]; must sum along the curve (length is always positive)

When To Use

Finding length of a curve given as y = f(x); differentiate f(x) to get dy/dx

Formula

L = ∫_c^d √[1 + (dx/dy)²] dy (Arc Length: curve x = g(y) from y = c to y = d)

Meaning

dx/dy = g'(y); integrate along y-axis

Watch Out

√[1 + (dx/dy)²]; same structure, different variable

When To Use

When curve is naturally expressed as x = g(y), or when dx/dy is simpler than dy/dx

Formula

L = ∫_a^b √[x'(t)² + y'(t)²] dt (Arc Length: parametric curve x = x(t), y = y(t), t ∈ [a,b])

Meaning

x'(t) = dx/dt; y'(t) = dy/dt; integrand is speed along curve

Watch Out

Both x'² and y'² appear under square root (not subtracted); t bounds, NOT x bounds

When To Use

Parametric curves (e.g., projectile motion, circle parametrization); avoids explicit y = f(x)

Term

Arc Length

Example

Length of y = x^(3/2) from x = 0 to x = 1: L = ∫_0^1 √[1 + (3x/2)²] dx

Definition

Total distance measured along a curve between two points; always positive; computed by integrating the differential arc length ds.

Values

• Single radius R from axis to curve; stacked perpendicular to axis
• Single curve; solid region (no hole)
• V = π ∫ R² da
• x-axis, y-axis

Property

Disk

Values

• Two radii: R_outer (top) and R_inner (bottom); stacked perpendicular to axis
• Region between two curves; creates annular (ring) cross-section
• V = π ∫ (R_outer² − R_inner²) da
• x-axis, y-axis

Property

Washer

Values

• Cylindrical shells with radius r, height h, thickness dr (or dy); summed radially
• Any curve; especially useful if axis of revolution not aligned with natural strip orientation
• V = 2π ∫ r·h da
• y-axis (x ∫), x-axis (y ∫)

Property

Shell

Values

• dA = f(x) dx
• dA = g(y) dy
• A = Σ A_i

Property

Area Element

Values

• M_y = ∫ x·f(x) dx
• M_y = ∫ x·g(y) dy
• M_y = Σ (x̄_i · A_i)

Property

First Moment M_y

Values

• M_x = ∫ (1/2)[f(x)]² dx
• M_x = ∫ y·g(y) dy
• M_x = Σ (ȳ_i · A_i)

Property

First Moment M_x

Values

• x̄ = M_y/A
• x̄ = M_y/A
• x̄ = Σ(x̄_i·A_i) / Σ A_i

Property

Centroid x̄

Values

• ȳ = M_x/A
• ȳ = M_x/A
• ȳ = Σ(ȳ_i·A_i) / Σ A_i

Property

Centroid ȳ

Values

• Composite functions (chain rule reverse)
• Let u = g(x), du = g'(x) dx, substitute, integrate ∫ f(u) du, back-substitute
• du MUST match integrand; don't forget back-substitution; check definite integral bounds change

Property

U-Substitution

Values

• Products: x·sin x, x·e^x, ln x·(poly)
• ∫ u dv = uv − ∫ v du; LIATE rule: u = Logarithm, Inverse trig, Algebra, Trig, Exponential (top priority)
• Wrong u choice leads to harder integral; may cycle (apply parts twice); watch signs; ∫ v du can be as hard as original

Property

Integration by Parts

Values

• Rational functions P(x)/Q(x), deg(P) < deg(Q)
• Factor Q(x); decompose into A/(x−a) + B/(x−b) + ... ; match coefficients or substitute convenient x values
• If deg(P) ≥ deg(Q), do long division first; repeated factors: (x−a)² requires both A/(x−a) and B/(x−a)²

Property

Partial Fractions

Values

• x^(n+1)/(n+1) + C
• n ≠ −1; add 1 to exponent, divide by new exponent

Property

x^n

Values

• ln|x| + C
• Absolute value essential; domain: x ≠ 0

Property

1/x

Values

• e^x + C
• No coefficient; e^(kx) → (1/k)e^(kx) + C

Property

e^x

Values

• a^x / ln(a) + C
• a > 0, a ≠ 1; denominator is ln(a), NOT log(a)

Property

a^x

Values

• −cos x + C
• Negative sign; sin(kx) → −(1/k)cos(kx) + C

Property

sin x

Values

• sin x + C
• Positive sign; cos(kx) → (1/k)sin(kx) + C

Property

cos x

Values

• −ln|cos x| + C OR ln|sec x| + C
• Either form equivalent

Property

tan x

Values

• tan x + C
• Inverse of tan; sec^2(kx) → (1/k)tan(kx) + C

Property

sec^2 x

Values

• −cot x + C
• Negative sign; csc^2(kx) → −(1/k)cot(kx) + C

Property

csc^2 x

Values

• sec x + C
• sec(kx)tan(kx) → (1/k)sec(kx) + C

Property

sec x tan x

Values

• arcsin x + C
• Domain: |x| < 1; ∫ 1/√(a²−x²) dx = arcsin(x/a) + C

Property

1/√(1−x²)

Values

• arctan x + C
• Domain: all x; ∫ 1/(a²+x²) dx = (1/a)arctan(x/a) + C

Property

1/(1+x²)