Definite Integration — JEE Mains Mathematics

Formulas (4)

• Substitution: tan x, tan(x/2) and rationalising · Half-angle substitution
  $$t=\tan\frac x2:\quad \sin x=\frac{2t}{1+t^2},\ \cos x=\frac{1-t^2}{1+t^2}$$
• Parts, partial fractions and inverse-tangent splits · By parts
  $$\int_a^b u\,v'\,dx=\big[uv\big]_a^b-\int_a^b u'\,v\,dx$$
• Find the integrand first, then integrate · Fourth power of cosine
  $$\cos^4x=\frac38+\frac12\cos2x+\frac18\cos4x$$
• Bounding an integral without evaluating it · Bounds from the extreme values
  $$m(b-a)\le\int_a^b f(x)\,dx\le M(b-a)$$

Watch out for (4)

• The limits change too
  → Substitution: tan x, tan(x/2) and rationalising
• Work out the boundary term
  → Parts, partial fractions and inverse-tangent splits
• Check the function you found
  → Find the integrand first, then integrate
• Check the bound against the options
  → Bounding an integral without evaluating it

Formulas (2)

• The a + b − x property · Reflection in the midpoint
  $$\int_a^b f(x)\,dx=\int_a^b f(a+b-x)\,dx$$
• The x → 1/x substitution · Reciprocal substitution
  $$\int_{1/a}^{a} f(x)\,dx=\int_{1/a}^{a} f\!\left(\tfrac1x\right)\frac{dx}{x^2}$$

Watch out for (2)

• The sum must be simpler
  → The a + b − x property
• Only for positive x
  → The x → 1/x substitution

Formulas (3)

• Odd parts vanish, even parts double · Symmetric limits
  $$\int_{-a}^{a}f=\begin{cases}0,&f\text{ odd}\\2\int_0^af,&f\text{ even}\end{cases}$$
• A denominator 1 + bᵍ⁽ˣ⁾ with g odd · Halving an even integrand
  $$\int_{-a}^{a}\frac{h(x)}{1+b^{g(x)}}dx=\int_0^{a}h(x)\,dx$$
• Periodic integrands · Many periods
  $$\int_0^{nT}f(x)\,dx=n\int_0^{T}f(x)\,dx$$

Watch out for (3)

• Check the whole term
  → Odd parts vanish, even parts double
• The numerator must be even
  → A denominator 1 + bᵍ⁽ˣ⁾ with g odd
• The right period
  → Periodic integrands

Formulas (3)

• Greatest integer: split where the value jumps · Greatest-integer integral
  $$\int_a^b[g(x)]\,dx=\sum_k k\cdot\ell\{x: [g(x)]=k\}$$
• Fractional part and repeating pieces · Whole periods of the fractional part
  $$\int_0^{n} g(\{x\})\,dx=n\int_0^{1} g(x)\,dx$$
• Modulus and the larger or smaller of two functions · Modulus split at the roots
  $$\int_a^b|g(x)|\,dx=\sum_{\text{pieces}}\left|\int g(x)\,dx\right|$$

Watch out for (3)

• Negative numbers round down
  → Greatest integer: split where the value jumps
• The partial period at the end
  → Fractional part and repeating pieces
• Every root in the interval
  → Modulus and the larger or smaller of two functions

Formulas (3)

• Differentiating an integral with variable limits · Leibniz's rule
  $$\frac{d}{dx}\int_{u(x)}^{v(x)}f(t)\,dt=f\big(v(x)\big)v'(x)-f\big(u(x)\big)u'(x)$$
• Definite integrals as unknown constants · The form it forces
  $$f(x)=g(x)+A\,h(x),\qquad A=\int_a^b k(t)f(t)\,dt$$
• Integrals of f(λx): power-function solutions · Power functions
  $$f(x)=cx^k\ \Rightarrow\ \int_0^1 f(\lambda x)\,d\lambda=\frac{f(x)}{k+1}$$

Watch out for (3)

• The chain factor
  → Differentiating an integral with variable limits
• x inside the integral is not a constant
  → Definite integrals as unknown constants
• Confirm the guess
  → Integrals of f(λx): power-function solutions

Formulas (2)

• Reduction formulas by parts · Powers of sine
  $$\int_0^{\pi/2}\sin^n x\,dx=\frac{n-1}{n}\int_0^{\pi/2}\sin^{n-2}x\,dx$$
• Beta integrals · The sum rule
  $$B(m+1,n)+B(m,n+1)=B(m,n)$$

Watch out for (2)

• Keep the boundary term
  → Reduction formulas by parts
• The exponents are m − 1 and n − 1
  → Beta integrals

Formulas (1)

• Sums as integrals · Riemann sum
  $$\lim_{n\to\infty}\frac1n\sum_{k=1}^{n}f\!\left(\frac kn\right)=\int_0^1 f(x)\,dx$$

Watch out for (1)

• Count the terms
  → Sums as integrals