Definite Integration formulas

Every formula JEE Advanced expects you to know for Definite Integration, with the variables spelled out. Free, no sign-up. Read it before a mock, not during one.

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Formulas: 32
Subject: Mathematics
Exam: JEE Advanced
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Definite Integration, every formula

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Fundamental theorem
$\int_{a}^{b} f (x) d x = F (b) - F (a)$
F'=f
Equal limits
$\int_{a}^{a} f (x) d x = 0$
Reversal of limits
$\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x$
Additivity
$\int_{a}^{b} f d x = \int_{a}^{c} f d x + \int_{c}^{b} f d x$
King's rule (general)
$\int_{a}^{b} f (x) d x = \int_{a}^{b} f (a + b - x) d x$
King's rule (0 to a)
$\int_{0}^{a} f (x) d x = \int_{0}^{a} f (a - x) d x$
Even/odd over [-a,a]
$\int_{- a}^{a} f d x = {2 \int_{0}^{a} f d x 0 f even f odd$
Zero to 2a rule
$\int_{0}^{2 a} f d x = {2 \int_{0}^{a} f d x 0 f (2 a - x) = f (x) f (2 a - x) = - f (x)$
Periodic function
$\int_{0}^{n T} f (x) d x = n \int_{0}^{T} f (x) d x$
T = period, n∈N
Leibnitz rule
$\frac{d}{d x} \int_{ϕ (x)}^{ψ (x)} f (t) d t = f (ψ) ψ^{'} - f (ϕ) ϕ^{'}$
Bounds of an integral
$m (b - a) \leq \int_{a}^{b} f d x \leq M (b - a)$
m,M = min,max of f on [a,b]
Modulus inequality
$\int_{a}^{b} f d x \leq \int_{a}^{b} ∣ f ∣ d x$
Mean value of a function
$\overset{ˉ}{f} = \frac{1}{b - a} \int_{a}^{b} f (x) d x$
Wallis symmetry
$\int_{0}^{π /2} sin^{n} x d x = \int_{0}^{π /2} cos^{n} x d x$
Reduction (even power)
$\int_{0}^{π /2} sin^{n} x d x = \frac{n - 1}{n} \cdot \frac{n - 3}{n - 2} \dots \frac{1}{2} \cdot \frac{π}{2}$
n even
Reduction (odd power)
$\int_{0}^{π /2} sin^{n} x d x = \frac{n - 1}{n} \cdot \frac{n - 3}{n - 2} \dots \frac{2}{3}$
n odd
Log-sine integral
$\int_{0}^{π /2} ln (sin x) d x = - \frac{π}{2} ln 2$
Log(1+tan) integral
$\int_{0}^{π /4} ln (1 + tan x) d x = \frac{π}{8} ln 2$
f(sin)/[f(sin)+f(cos)]
$\int_{0}^{π /2} \frac{f ( sin x )}{f ( sin x ) + f ( cos x )} d x = \frac{π}{4}$
Reciprocal quadratic in sin/cos
$\int_{0}^{π /2} \frac{d x}{a ^{2} sin ^{2} x + b ^{2} cos ^{2} x} = \frac{π}{2 ab}$
a,b>0
Laplace-type cosine
$\int_{0}^{\infty} e^{- a x} cos b x d x = \frac{a}{a ^{2} + b ^{2}}$
a>0
Laplace-type sine
$\int_{0}^{\infty} e^{- a x} sin b x d x = \frac{b}{a ^{2} + b ^{2}}$
a>0
Beta integral
$\int_{0}^{1} x^{m} (1 - x)^{n} d x = \frac{m ! n !}{( m + n + 1 )!}$
m,n∈N
Square-root product
$\int_{a}^{b} (x - a) (b - x) d x = \frac{π}{8} (b - a)^{2}$
Reciprocal square-root product
$\int_{a}^{b} \frac{d x}{( x - a ) ( b - x )} = π$
Integral of greatest integer
$\int_{0}^{n} ⌊ x ⌋ d x = \frac{n ( n - 1 )}{2}$
n∈N
Integral of fractional part
$\int_{0}^{n} {x} d x = \frac{n}{2}$
n∈N
x·f(sin x) over [0,π]
$\int_{0}^{π} x f (sin x) d x = \frac{π}{2} \int_{0}^{π} f (sin x) d x$
Tangent reduction
$\int_{0}^{π /4} (tan^{n} x + tan^{n - 2} x) d x = \frac{1}{n - 1}$
Integral as a limit of a sum
$\int_{0}^{1} f (x) d x = n \to \infty lim \frac{1}{n} r = 1 \sum n f (\frac{r}{n})$
Log-cosine integral
$\int_{0}^{π /2} ln (cos x) d x = - \frac{π}{2} ln 2$
Comparison of integrals
$f (x) \leq g (x) on [a, b] \Rightarrow \int_{a}^{b} f d x \leq \int_{a}^{b} g d x$

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Definite Integration formulas

Definite Integration, every formula

Fundamental theorem

Equal limits

Reversal of limits

Additivity

King's rule (general)

King's rule (0 to a)

Even/odd over [-a,a]

Zero to 2a rule

Periodic function

Leibnitz rule

Bounds of an integral

Modulus inequality

Mean value of a function

Wallis symmetry

Reduction (even power)

Reduction (odd power)

Log-sine integral

Log(1+tan) integral

f(sin)/[f(sin)+f(cos)]

Reciprocal quadratic in sin/cos

Laplace-type cosine

Laplace-type sine

Beta integral

Square-root product

Reciprocal square-root product

Integral of greatest integer

Integral of fractional part

x·f(sin x) over [0,π]

Tangent reduction

Integral as a limit of a sum

Log-cosine integral

Comparison of integrals

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