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Definite Integration formulas

18 formulas and 18 common traps for JEE Mains Maths Definite Integration, grouped by subtopic.

Full notes with worked examples

Evaluating by Substitution, Parts and Partial Fractions

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Substitution: tan x, tan(x/2) and rationalising

Half-angle substitution

t=tan⁡x2:sin⁡x=2t1+t2, cos⁡x=1−t21+t2t=\tan\frac x2:\quad \sin x=\frac{2t}{1+t^2},\ \cos x=\frac{1-t^2}{1+t^2}

Parts, partial fractions and inverse-tangent splits

By parts

∫abu v′ dx=[uv]ab−∫abu′ v dx\int_a^b u\,v'\,dx=\big[uv\big]_a^b-\int_a^b u'\,v\,dx

Find the integrand first, then integrate

Fourth power of cosine

cos⁡4x=38+12cos⁡2x+18cos⁡4x\cos^4x=\frac38+\frac12\cos2x+\frac18\cos4x

Bounding an integral without evaluating it

Bounds from the extreme values

m(b−a)≤∫abf(x) dx≤M(b−a)m(b-a)\le\int_a^b f(x)\,dx\le M(b-a)

Common traps

The limits change too

With t=tan⁡xt=\tan x, the limit x=π2x=\frac\pi2 becomes t→∞t\to\infty, and with t=tan⁡x2t=\tan\frac x2 it becomes t=1t=1. Keeping the old limits is the most common slip.

Work out the boundary term

[uv]ab[uv]_a^b is a number, not zero by default. It is zero only when uvuv vanishes at both limits; check before dropping it.

Check the function you found

A function recovered from one condition may fail another. Test it against every given value before integrating.

Check the bound against the options

Work out the two bounds as decimals and compare with each option. If the bounds fit no option, recheck the monotonicity before trusting either.

The a + b − x Property and Other Symmetries

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The a + b − x property

Reflection in the midpoint

∫abf(x) dx=∫abf(a+b−x) dx\int_a^b f(x)\,dx=\int_a^b f(a+b-x)\,dx

The x → 1/x substitution

Reciprocal substitution

∫1/aaf(x) dx=∫1/aaf ⁣(1x)dxx2\int_{1/a}^{a} f(x)\,dx=\int_{1/a}^{a} f\!\left(\tfrac1x\right)\frac{dx}{x^2}

Common traps

The sum must be simpler

The property always holds, but it helps only when the two forms add to something easier. If they do not, the question needs a different method.

Only for positive x

tan⁡−1x+tan⁡−11x=π2\tan^{-1}x+\tan^{-1}\frac1x=\frac\pi2 holds for x>0x>0; for x<0x<0 the sum is −π2-\frac\pi2.

Odd, Even and Periodic Integrands

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Odd parts vanish, even parts double

Symmetric limits

∫−aaf={0,f odd2∫0af,f even\int_{-a}^{a}f=\begin{cases}0,&f\text{ odd}\\2\int_0^af,&f\text{ even}\end{cases}

A denominator 1 + bᵍ⁽ˣ⁾ with g odd

Halving an even integrand

∫−aah(x)1+bg(x)dx=∫0ah(x) dx\int_{-a}^{a}\frac{h(x)}{1+b^{g(x)}}dx=\int_0^{a}h(x)\,dx

Periodic integrands

Many periods

∫0nTf(x) dx=n∫0Tf(x) dx\int_0^{nT}f(x)\,dx=n\int_0^{T}f(x)\,dx

Common traps

Check the whole term

x∣x∣x|x| is odd but x2∣x∣x^2|x| is even. Decide the parity of each complete term, not of its pieces.

The numerator must be even

An odd numerator over 1+bg1+b^{g} does not halve. Split the numerator into even and odd parts and treat each by its own rule.

The right period

∣sin⁡x∣|\sin x| repeats every π\pi, not 2π2\pi. Using the longer period halves the number of copies and halves the answer.

Greatest Integer, Modulus and Max–Min Integrands

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Greatest integer: split where the value jumps

Greatest-integer integral

∫ab[g(x)] dx=∑kk⋅ℓ{x:[g(x)]=k}\int_a^b[g(x)]\,dx=\sum_k k\cdot\ell\{x: [g(x)]=k\}

Fractional part and repeating pieces

Whole periods of the fractional part

∫0ng({x}) dx=n∫01g(x) dx\int_0^{n} g(\{x\})\,dx=n\int_0^{1} g(x)\,dx

Modulus and the larger or smaller of two functions

Modulus split at the roots

∫ab∣g(x)∣ dx=∑pieces∣∫g(x) dx∣\int_a^b|g(x)|\,dx=\sum_{\text{pieces}}\left|\int g(x)\,dx\right|

Common traps

Negative numbers round down

[−0.3]=−1[-0.3]=-1, not 0. On an interval below zero, [x][x] is the next integer to the left.

The partial period at the end

With an upper limit like 10.5, there are ten whole periods and half of one more. Add the half-period's integral; it is not half of a full one unless the integrand is constant.

Every root in the interval

A quadratic may have two roots inside the limits. Missing one leaves a piece with the wrong sign, and the error is exactly twice that piece's integral.

Integral Equations and Leibniz's Rule

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Differentiating an integral with variable limits

Leibniz's rule

ddx∫u(x)v(x)f(t) dt=f(v(x))v′(x)−f(u(x))u′(x)\frac{d}{dx}\int_{u(x)}^{v(x)}f(t)\,dt=f\big(v(x)\big)v'(x)-f\big(u(x)\big)u'(x)

Definite integrals as unknown constants

The form it forces

f(x)=g(x)+A h(x),A=∫abk(t)f(t) dtf(x)=g(x)+A\,h(x),\qquad A=\int_a^b k(t)f(t)\,dt

Integrals of f(λx): power-function solutions

Power functions

f(x)=cxk ⇒ ∫01f(λx) dλ=f(x)k+1f(x)=cx^k\ \Rightarrow\ \int_0^1 f(\lambda x)\,d\lambda=\frac{f(x)}{k+1}

Common traps

The chain factor

An upper limit x2x^2 brings a factor 2x2x; a lower limit contributes with a minus sign. Forgetting either is the usual slip.

x inside the integral is not a constant

In ∫01(x−t)f(t) dt\int_0^1(x-t)f(t)\,dt the xx must come out first: it is x∫f−∫tfx\int f-\int tf. Naming the whole integral a constant is wrong.

Confirm the guess

A power function is a guess. Check that it satisfies the equation for every xx, and that the value of aa it forces matches the one given.

Reduction Formulas and Beta Integrals

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Reduction formulas by parts

Powers of sine

∫0π/2sin⁡nx dx=n−1n∫0π/2sin⁡n−2x dx\int_0^{\pi/2}\sin^n x\,dx=\frac{n-1}{n}\int_0^{\pi/2}\sin^{n-2}x\,dx

Beta integrals

The sum rule

B(m+1,n)+B(m,n+1)=B(m,n)B(m+1,n)+B(m,n+1)=B(m,n)

Common traps

Keep the boundary term

The term [uv][uv] from integrating by parts is often what gives the relation its constant, as the ee in In=e−nIn−1I_n=e-nI_{n-1}. Dropping it makes the relation wrong.

The exponents are m − 1 and n − 1

x2(1−x)2x^2(1-x)^2 is B(3,3)B(3,3), not B(2,2)B(2,2). Add 1 to each exponent to read off mm and nn.

Limits of Sums as Integrals

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Sums as integrals

Riemann sum

lim⁡n→∞1n∑k=1nf ⁣(kn)=∫01f(x) dx\lim_{n\to\infty}\frac1n\sum_{k=1}^{n}f\!\left(\frac kn\right)=\int_0^1 f(x)\,dx

Common traps

Count the terms

The range of kk sets the upper limit. A sum to 2n2n or 3n3n integrates to 2 or 3, not 1.

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