Definite Integration — NDA Mathematics Formulas and Traps

Fundamental Theorem, Periodicity and the Leibniz Rule

The Fundamental Theorem of Calculus

Fundamental Theorem of Calculus and its corollaries

\int_a^b f(x)\,dx = F(b)-F(a)\;\;(F'=f) \qquad \int_a^b f'(x)\,dx = f(b)-f(a) \qquad \int \frac{f'(x)}{f(x)}\,dx = \ln|f(x)|

Integrals of periodic functions

Integral of a periodic function

\int_0^{nT} f(x)\,dx = n\int_0^{T} f(x)\,dx \qquad \int_{a}^{a+nT} f(x)\,dx = n\int_0^{T} f(x)\,dx

The Leibniz rule — differentiating an integral

Leibniz rule (variable upper limit)

\frac{d}{dx}\int_{a}^{g(x)} f(t)\,dt = f(g(x))\,g'(x)

Common traps

Integrating a derivative is not always trivial

\int_{-1}^{1}\frac{d}{dx}\big(\tan^{-1}\frac1x\big)\,dx

is NOT

[\tan^{-1}\frac1x]_{-1}^1

naively, because the function jumps at

x=0

. Compute the derivative

\frac{-1}{1+x^2}

first, then integrate to get

-\frac{\pi}{2}

.

Use the function's period, not the trig argument's

\sin^4x+\cos^4x

does NOT have period

2\pi

— squaring and adding shrinks the period to

\tfrac{\pi}{2}

. Always determine the actual period of the whole integrand before counting how many fit.

Don't forget the chain-rule factor

\frac{d}{dx}\int_a^{g(x)} f

is

f(g(x))\cdot g'(x)

, NOT just

f(g(x))

. The

g'(x)

factor is the most commonly dropped term.

Properties — King's, Symmetry and Standard Results

Learn this subtopic in the notes

King's property — the reflection trick

King's property and its companions

\int_0^a f(x)\,dx = \int_0^a f(a-x)\,dx \qquad \int_a^b f(x)\,dx = \int_a^b f(a+b-x)\,dx \qquad \int_0^{2a} f(x)\,dx = \int_0^a \big[f(x)+f(2a-x)\big]\,dx \qquad \int_0^a \frac{f(x)}{f(x)+f(a-x)}\,dx = \frac{a}{2}

Odd/even symmetry over a symmetric interval

\int_{-a}^{a} f(x)\,dx = 0 \;\;(f\text{ odd}) \qquad \int_{-a}^{a} f(x)\,dx = 2\int_0^a f(x)\,dx \;\;(f\text{ even}) \qquad \int_{-a}^{a} \frac{f(x)}{1+c^{x}}\,dx = \int_0^a f(x)\,dx \;\;(f\text{ even})

Standard results and trig reductions

Standard definite-integral results

\int_0^{\pi}\frac{dx}{1+\sin^2x} = \frac{\pi}{\sqrt2} \qquad \int_0^1 x^{m}(1-x)^{n}\,dx = \frac{m!\,n!}{(m+n+1)!} \qquad \int_0^{\pi/2}\sin^{2}x\,dx = \int_0^{\pi/2}\cos^{2}x\,dx = \frac{\pi}{4}

Common traps

Add the reflected form — don't just substitute and stop

King's property by itself only rewrites

I

; the magic is ADDING the original and the reflected integral so the numerator becomes the denominator. If you substitute and forget to add, you go in a circle.

King's reflection only works with the matching limits

The reflection is

x\to a+b-x

for limits

[a,b]

— NOT a fixed

x\to a-x

. On

\int_2^5 f\,dx

you must replace

x

by

7-x

, not

2-x

. Using the wrong reflection sends the limits outside the interval and the cancellation fails.

Check parity of the WHOLE integrand

A sum can have one odd part and one non-odd part. Split it: the odd piece dies, but the rest must still be integrated. Don't declare the whole integral zero just because one term is odd.

Reduce the power BEFORE integrating

Don't integrate

\sin^4x

and

\cos^4x

separately — combine them via

\sin^4x+\cos^4x=\frac{3+\cos4x}{4}

first. The reduction turns a tedious computation into a one-line answer.

Transform the limits when you substitute

When you substitute

u=g(x)

in a definite integral, change the limits to

g(a)

and

g(b)

and you never need to back-substitute. Forgetting to convert the limits is the classic definite-integral error.

Absolute Value, Piecewise and Greatest-Integer Integrals

Learn this subtopic in the notes

Integrating greatest-integer (floor) functions

Greatest-integer (floor) results

\int_n^{n+1}\big(x-\lfloor x\rfloor\big)\,dx = \frac12 \qquad \lfloor x\rfloor + \lfloor -x\rfloor = -1 \;\;(x\notin\mathbb{Z}) \qquad \int_0^{n}\lfloor x\rfloor\,dx = \frac{n(n-1)}{2}

Common traps

Find the zeros first — don't drop the absolute value

\int_{-1}^{1}(1-x^2)\,dx

(no bars) gives a SIGNED area, but

\int_{-1}^{1}|x^2-1|\,dx

needs the sign of

x^2-1

on each piece. Forgetting the bars (or the split) gives the wrong, signed value.

Split |x| at 0, even when 0 is interior to the interval

For

\int_{-2}^{3}|x|\,dx

you cannot just use

\tfrac{x^2}{2}

across the whole range:

|x|=-x

on

[-2,0]

and

|x|=x

on

[0,3]

. Split at the break-point inside the interval and add

2 + \tfrac{9}{2} = \tfrac{13}{2}

; skipping the split gives the wrong

\tfrac{5}{2}

.

⌊x⌋ on [−1, 0) is −1, not 0

The floor of a negative non-integer rounds DOWN:

\lfloor -0.3\rfloor = -1

. A common slip is using

\lfloor x\rfloor=0

on

[-1,0)

— it is

-1

there, which flips the sign of the contribution.

Area Under and Between Curves

Learn this subtopic in the notes

Area between curves

Area as a definite integral

A = \int_a^b |f(x)|\,dx \qquad A = \int_a^b |f(x)-g(x)|\,dx

Common traps

Area is unsigned — use the absolute value

Plain

\int_{-1}^{1}(x^2-1)\,dx

gives a NEGATIVE number, which cannot be an area. Take

|f|

(or split at the roots and add magnitudes) so the region below the axis still adds positively.

Recovering a Function from Integral Conditions

Learn this subtopic in the notes

Common traps

One condition, one equation — match the counts

You need as many independent conditions as unknown parameters. With three unknowns P, Q, R you must extract three equations (e.g. a value, an integral, and a derivative) before the system is solvable.

Definite Integration formulas

Fundamental Theorem, Periodicity and the Leibniz Rule

The Fundamental Theorem of Calculus

Integrals of periodic functions

The Leibniz rule — differentiating an integral

Common traps

Properties — King's, Symmetry and Standard Results

King's property — the reflection trick

Standard results and trig reductions

Common traps

Absolute Value, Piecewise and Greatest-Integer Integrals

Integrating greatest-integer (floor) functions

Common traps

Area Under and Between Curves

Area between curves

Common traps

Recovering a Function from Integral Conditions

Common traps

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