Substitution method — common mistake

“Skipping the new coordinates after rotating a map is like forgetting to change limits in a definite integral.”

When a definite integral is evaluated by substitution, the limits must be rewritten in the new variable u. Using the original x-limits after the change gives an incorrect numerical value even if the antiderivative is right. Always transform limits together with the integrand.

Memory trick: Limits travel with u.

Integration by parts

“Like a chef separating egg whites from yolk before whisking, we split the integrand into u and dv.”

∫u dv = uv − ∫v du. Pick u so that du simplifies and dv is easily integrable; ILATE guides this. A trap: if the new integral repeats the original, bring it to the left and solve algebraically.

Memory trick: If integral repeats, move it to LHS and solve.

Integration by parts — common mistake

“Choosing u as sin x instead of cos x is like lifting the heavier suitcase first.”

Wrong u makes du harder and dv harder to integrate, increasing work. For polynomial·exp, always let the polynomial be u. Trap: never differentiate a function you intended to integrate as dv.

Memory trick: Poly → differentiate, exp → integrate.

Partial fractions

“Breaking a complex circuit into simple resistors mirrors splitting a rational function into partial fractions.”

A proper rational function (deg num < deg den) can be expressed as sum of A/(linear) or (Bx+C)/(quadratic). Ensure denominator is fully factored over reals. Trap: forgetting repeated factors leads to missing terms.

Memory trick: Degree(num) < degree(den) before split.

Partial fractions — common mistake

“Skipping the degree check is like trying to pour a larger bucket into a smaller one.”

If deg num ≥ deg den, first perform polynomial long division to obtain a polynomial plus a proper fraction. Ignoring this gives wrong A,B constants. Trap: forgetting the constant term for quadratic denominators.

Memory trick: Long-divide first when numerator degree ≥ denominator.

Properties of definite integrals

“A seesaw balanced at the centre shows how odd functions cancel over symmetric limits.”

For f(−x)=−f(x), ∫_(−a)ᵃ f(x)dx =0; for even f, integral doubles from 0 to a. The king-property: ∫ₐᵇ f(x)dx = ∫ₐᵇ f(a+b−x)dx. Trap: applying odd rule when limits aren’t symmetric about zero.

Memory trick: Odd+[-a,a]=0; Even+[-a,a]=2∫0ᵃ.

Properties of definite integrals — common mistake

“Missing symmetry is like ignoring a mirror; the integral won’t cancel.”

The common error is using odd-function zero rule on limits like 0 to π. Symmetry must be about the origin; a shift destroys cancellation. Always verify limits before applying parity.

Memory trick: Odd zero only for limits −a to a.

Fundamental theorem of calculus

“Watching a car’s odometer from start a to finish b shows the change equals F(b)−F(a), where F is the distance-vs-time antiderivative.”

FTC I: if f is continuous on [a,b] and F′=f, then ∫_aᵇ f(x)dx = F(b)−F(a). It yields net signed area, not total distance unless f≥0. Common trap: swapping limits (F(a)−F(b)) flips the sign, and applying it when f has a jump discontinuity is invalid.

Memory trick: Upper-minus-lower → F(b)−F(a)

Fundamental theorem of calculus — common mistake

“Reading a recipe backward swaps the first and last steps, just as swapping upper and lower limits flips the sign of the integral.”

The definite integral equals F(b)−F(a). Writing F(a)−F(b) gives the negative answer. This sign error is common when changing variables; always adjust limits consistently.

Memory trick: Upper minus lower, never reverse

Integral of x^n = x^(n+1)/(n+1) + C

“Stacking blocks where each layer’s height follows xⁿ adds one more exponent, just like the power rule adds 1 to n.”

∫xⁿdx = xⁿ⁺¹/(n+1)+C for n≠−1. The extra ‘+1’ comes from reversing differentiation. Forgetting n≠−1 leads to division by zero; the correct case uses the logarithm rule.

Integral of xⁿ = x^(n+1)/(n+1) + C

Memory trick: n≠−1 → raise power, divide by new exponent

Integral of 1/x = ln|x| + C

“Seismologists compute earthquake magnitude as log10|amplitude|; the absolute value mirrors ln|x| in ∫1/x dx.”

∫1/x dx = ln|x|+C for any x≠0. The absolute value makes the derivative 1/x both for positive and negative x. If a definite integral’s limits straddle 0 the integral is improper and diverges; you cannot just substitute endpoints in ln|x|. Common trap: dropping the bars gives a wrong sign for negative limits.

Integral of 1/x = ln|x| + C

Memory trick: ln|x| → remember the bars: ‘absolute’ keeps derivative 1/x on both sides.

Integration by parts: u v - integral v du

“In a dance, the leader (u) moves first, the partner (dv) follows, and the final pose subtracts the leftover steps: u·v−∫v·du.”

Integration by parts: ∫u dv = u·v − ∫v du. Choose u by ILATE (Inverse trig, Log, Algebra, Trig, Exp). A trap: forgetting to differentiate u correctly or to keep the minus sign after integration flips the result.

Integration by parts: u v - integral v du

Memory trick: IBP: u·v−∫v·du, pick u by ILATE

Integral_a^b f = F(b) - F(a)

“Measuring water flow from valve A to B is like computing ∫_aᵇ f, you subtract the level at A from that at B.”

If F′=f on [a,b] and F is continuous, ∫_aᵇ f(x)dx = F(b)−F(a). When f has a discontinuity inside, split the integral at that point; otherwise FTC gives wrong net area. Add C only for indefinite integrals.

Integral_aᵇ f = F(b) - F(a)

Memory trick: Definite = F(b)−F(a); split at breaks

Integral_0^a f(x) dx = Integral_0^a f(a-x) dx

“Swapping the left and right halves of a chocolate bar of length a leaves the total taste unchanged.”

Using u = a−x, dx = −du flips the limits, so the integral of f(x) from 0 to a equals that of f(a−x). It holds for any integrable f. The common trap is forgetting the sign change of dx; the limits swap and cancel the minus.

Integral₀ᵃ f(x) dx = Integral₀ᵃ f(a-x) dx

Memory trick: Flip the interval, same area

Odd function over [-a, a] integrates to 0

“A perfectly balanced seesaw has equal weights at opposite distances, giving zero net torque.”

If f(−x)=−f(x) (odd), the area left of the origin cancels the right, so ∫_(−a)ᵃf(x)dx=0. This works only for symmetric limits; shifting the interval destroys the cancellation. Watch out for non-symmetric bounds.

Odd function over [-a, a] integrates to 0

Memory trick: Odd → cancel

definite integral vs indefinite integral

“Measuring water in a tank gives a litre count; writing a recipe tells how to fill it later.”

A definite integral has limits and yields a single number (signed area). An indefinite integral lacks limits and gives the antiderivative plus an arbitrary constant C. Forgetting +C in an initial-value problem loses the correct solution.

Memory trick: Limits → number; no limits → family + C

integration by substitution vs by parts

“Unwrapping a gift follows the inner pattern (substitution); separating mixed nuts needs picking each type (parts).”

Substitution reverses the chain rule when the integrand contains a function and its derivative. Integration by parts reverses the product rule, useful when differentiating one factor simplifies it. Applying parts without a simplifying derivative makes the integral harder.

Memory trick: Chain → sub, product → parts

area under a curve vs signed area

“Walking uphill then downhill gives zero net elevation change, but the total distance walked is the sum of both climbs.”

The definite integral gives signed area: portions below the x-axis count negative. To find total geometric area you must integrate |f(x)|. Exams often ask for area; forgetting the absolute value yields a wrong (possibly zero) answer.

Memory trick: Signed = net, absolute = total

Memory trick: Indefinite always + C

Order and degree

“Order is the tallest floor (highest derivative); degree is how many times that floor repeats (its exponent) once the blueprint has no curved (radical) lines.”

Order = highest order derivative present. Degree = exponent of that highest derivative after the DE is expressed as a polynomial in derivatives—no radicals, fractions, or transcendental functions of derivatives. Trap: reading degree before rationalising the equation gives a fractional or undefined degree.

Memory trick: Degree after polynomialising derivatives.

Order and degree — common mistake

“Don’t judge a DE’s degree before scraping off the frosting of radicals and fractions that hide the true layers of derivatives.”

Students often compute degree directly from the given form, overlooking radicals or fractional powers of derivatives. The correct procedure: first make the equation polynomial in all derivatives (clear radicals, denominators), then read the degree. Missing this step leads to wrong degree and marks lost.

Memory trick: Clear radicals/fractions → then read degree.

Variable separable

“Separating variables is like sorting red and blue marbles into two jars before counting them.”

If dy/dx = g(x)h(y), rewrite as dy/h(y)=g(x)dx and integrate both sides. Ensure each side contains only one variable; integrating before full separation misses terms or gives a wrong constant.

Memory trick: All x with dx, all y with dy.

Variable separable — common mistake

“Separating variables is like sorting red and blue balls before counting each colour.”

In a separable ODE, first rearrange so all x-terms with dx are on one side and all y-terms with dy on the other. Only then integrate each side; the integration constant is a true constant, not a hidden function. The classic trap is to integrate prematurely, which mixes the variables and gives an incorrect solution.

Memory trick: Separate → then integrate; S-then-I

Homogeneous equations

“Using y=vx is like riding a bike whose speed scales with the distance already covered.”

For a homogeneous ODE dy/dx = F(y/x), set y=vx. Then dy/dx = v + x dv/dx, giving a separable equation in v and x. Forgetting the extra v term is a common trap.

Memory trick: y=vx → dy/dx = v + x dv/dx.