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Definite Integration formulas

Master Definite Integration through 32 JEE Advanced-level formulas, systematically structured with every variable spelled out. Revise concept-wise, identify the areas where you need improvement, and focus your preparation with greater precision.

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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
$\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x$
Q1MCQReversal
$\displaystyle\int_3^1 f(x)\,dx$ equals:
- A
  $-\int_1^3 f\,dx$
- B
  $\int_1^3 f\,dx$
- C
  $0$
- D
  $2\int_1^3 f\,dx$
Additivity
$\int_{a}^{b} f d x = \int_{a}^{c} f d x + \int_{c}^{b} f d x$
Q1MCQAdditivity
$\displaystyle\int_0^{5} f\,dx$ equals:
- A
  $\int_0^{2} f\,dx+\int_2^{5} f\,dx$
- B
  $\int_0^{2} f\,dx-\int_2^{5} f\,dx$
- C
  $\int_2^{5} f\,dx$
- D
  $\int_0^{2} f\,dx$
King's rule (general)
$\int_{a}^{b} f (x) d x = \int_{a}^{b} f (a + b - x) d x$
Q1MCQKing's rule
$\displaystyle\int_a^b f(x)\,dx$ equals:
- A
  $\int_a^b f(a+b-x)\,dx$
- B
  $\int_a^b f(b-x)\,dx$
- C
  $\int_a^b f(a-x)\,dx$
- D
  $\int_a^b f(x-a)\,dx$
King's rule (0 to a)
$\int_{0}^{a} f (x) d x = \int_{0}^{a} f (a - x) d x$
Q1MCQKing's rule 0-a
$\displaystyle\int_0^a f(x)\,dx$ equals:
- A
  $\int_0^a f(a-x)\,dx$
- B
  $\int_0^a f(a+x)\,dx$
- C
  $\int_0^a f(-x)\,dx$
- D
  $2\int_0^a f\,dx$
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]
Q1MCQBounds
For $f$ on $[a,b]$ with min $m$, max $M$, the integral $\int_a^b f\,dx$ satisfies:
- A
  $m(b-a)\le\int_a^b f\le M(b-a)$
- B
  $\int_a^b f\ge M(b-a)$
- C
  $\int_a^b f\le m(b-a)$
- D
  $\int_a^b f=M(b-a)$
Modulus inequality
$\int_{a}^{b} f d x \leq \int_{a}^{b} ∣ f ∣ d x$
Q1MCQModulus inequality
$\left|\int_a^b f\,dx\right|$ compares with $\int_a^b|f|\,dx$ as:
- A
  $\le$
- B
  $\ge$
- C
  $=$ always
- D
  no relation
Mean value of a function
$\overset{ˉ}{f} = \frac{1}{b - a} \int_{a}^{b} f (x) d x$
Wallis symmetry
$\int_{0}^{π /2} sin^{n} x d x = \int_{0}^{π /2} cos^{n} x d x$
Q1MCQWallis symmetry
$\displaystyle\int_0^{\pi/2}\sin^{6}x\,dx$ equals:
- A
  $\int_0^{\pi/2}\cos^{6}x\,dx$
- B
  $\int_0^{\pi/2}\cos^{5}x\,dx$
- C
  $0$
- D
  $\int_0^{\pi}\sin^{6}x\,dx$
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
Q1MCQEven-power reduction
$\displaystyle\int_0^{\pi/2}\sin^{4}x\,dx$ equals:
- A
  $\dfrac{3\pi}{16}$
- B
  $\dfrac{\pi}{4}$
- C
  $\dfrac{3}{8}$
- D
  $\dfrac{\pi}{2}$
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
Q1MCQOdd-power reduction
$\displaystyle\int_0^{\pi/2}\sin^{3}x\,dx$ equals:
- A
  $\dfrac{2}{3}$
- B
  $\dfrac{3}{4}$
- C
  $\dfrac{\pi}{4}$
- D
  $\dfrac{1}{2}$
Log-sine integral
$\int_{0}^{π /2} ln (sin x) d x = - \frac{π}{2} ln 2$
Q1MCQLog-sine
$\displaystyle\int_0^{\pi/2}\ln(\sin x)\,dx$ equals:
- A
  $-\dfrac{\pi}{2}\ln 2$
- B
  $\dfrac{\pi}{2}\ln 2$
- C
  $-\dfrac{\pi}{4}\ln 2$
- D
  $0$
Log(1+tan) integral
$\int_{0}^{π /4} ln (1 + tan x) d x = \frac{π}{8} ln 2$
Q1MCQLog(1+tan)
$\displaystyle\int_0^{\pi/4}\ln(1+\tan x)\,dx$ equals:
- A
  $\dfrac{\pi}{8}\ln 2$
- B
  $\dfrac{\pi}{4}\ln 2$
- C
  $\dfrac{\pi}{2}\ln 2$
- D
  $\ln 2$
f(sin)/[f(sin)+f(cos)]
$\int_{0}^{π /2} \frac{f ( sin x )}{f ( sin x ) + f ( cos x )} d x = \frac{π}{4}$
Q1MCQSymmetric f(sin)
$\displaystyle\int_0^{\pi/2}\dfrac{\sin x}{\sin x+\cos x}\,dx$ equals:
- A
  $\dfrac{\pi}{4}$
- B
  $\dfrac{\pi}{2}$
- C
  $\dfrac{\pi}{8}$
- D
  $1$
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
Q1MCQQuadratic sin/cos
$\displaystyle\int_0^{\pi/2}\dfrac{dx}{\sin^{2}x+4\cos^{2}x}$ equals:
- A
  $\dfrac{\pi}{4}$
- B
  $\dfrac{\pi}{2}$
- C
  $\dfrac{\pi}{8}$
- D
  $\dfrac{\pi}{6}$
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
Q1MCQBeta integral
$\displaystyle\int_0^1 x^{2}(1-x)^{3}\,dx$ equals:
- A
  $\dfrac{1}{60}$
- B
  $\dfrac{1}{30}$
- C
  $\dfrac{1}{12}$
- D
  $\dfrac{1}{6}$
Square-root product
$\int_{a}^{b} (x - a) (b - x) d x = \frac{π}{8} (b - a)^{2}$
Q1MCQSquare-root product
$\displaystyle\int_0^{2}\sqrt{x(2-x)}\,dx$ equals:
- A
  $\dfrac{\pi}{2}$
- B
  $\pi$
- C
  $\dfrac{\pi}{4}$
- D
  $2$
Reciprocal square-root product
$\int_{a}^{b} \frac{d x}{( x - a ) ( b - x )} = π$
Q1MCQReciprocal root product
$\displaystyle\int_1^{5}\dfrac{dx}{\sqrt{(x-1)(5-x)}}$ equals:
- A
  $\pi$
- B
  $\dfrac{\pi}{2}$
- C
  $2\pi$
- D
  $\dfrac{\pi}{4}$
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]
$\int_{0}^{π} x f (sin x) d x = \frac{π}{2} \int_{0}^{π} f (sin x) d x$
Q1MCQx·f(sin x)
$\displaystyle\int_0^{\pi} x\,\sin x\,dx$ equals:
- A
  $\pi$
- B
  $2\pi$
- C
  $\dfrac{\pi}{2}$
- D
  $2$
Tangent reduction
$\int_{0}^{π /4} (tan^{n} x + tan^{n - 2} x) d x = \frac{1}{n - 1}$
Q1MCQTangent reduction
$\displaystyle\int_0^{\pi/4}(\tan^{4}x+\tan^{2}x)\,dx$ equals:
- A
  $\dfrac{1}{3}$
- B
  $\dfrac{1}{2}$
- C
  $\dfrac{1}{4}$
- D
  $1$
Integral as a limit of a sum
$\int_{0}^{1} f (x) d x = n \to \infty lim \frac{1}{n} r = 1 \sum n f (\frac{r}{n})$
Q1MCQLimit of a sum
$\displaystyle\lim_{n\to\infty}\sum_{r=1}^{n}\dfrac{1}{n+r}$ equals:
- A
  $\ln 2$
- B
  $1$
- C
  $\dfrac{1}{2}$
- D
  $\ln 3$
Log-cosine integral
$\int_{0}^{π /2} ln (cos x) d x = - \frac{π}{2} ln 2$
Q1MCQLog-cosine
$\displaystyle\int_0^{\pi/2}\ln(\cos x)\,dx$ equals:
- A
  $-\dfrac{\pi}{2}\ln 2$
- B
  $\dfrac{\pi}{2}\ln 2$
- C
  $0$
- D
  $-\dfrac{\pi}{4}\ln 2$
Comparison of integrals
$f (x) \leq g (x) on [a, b] \Rightarrow \int_{a}^{b} f d x \leq \int_{a}^{b} g d x$
Q1MCQComparison
If $f(x)\le g(x)$ on $[a,b]$, then:
- A
  $\int_a^b f\,dx\le\int_a^b g\,dx$
- B
  $\int_a^b f\,dx\ge\int_a^b g\,dx$
- C
  $\int_a^b f=\int_a^b g$
- D
  no relation

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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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