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Definite Integration formula sheet

Every formula in the Definite Integration chapter for JEE Main, with one line on when it applies. Topic 1 is open to everyone; sign in free to read all 29 and download the PDF.

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How we count: 7× asked means 7 past JEE Main questions in the PrepWiser bank needed that formula, and "last" is the latest year one appeared. ★ marks formulas asked 8 or more times.

Fundamental theorem and evaluation

Fundamental theorem
20× asked, last 2026
∫abf(x) dx=F(b)−F(a),F′=f on [a,b]\int_a^bf(x)\,dx=F(b)-F(a),\qquad F^{\prime}=f\text{ on }[a,b]
FF must not jump inside: −tan⁡−1(1/x)-\tan^{-1}(1/x) on [−1,1][-1,1] gives −π/2-\pi/2; the true value is π/2\pi/2. Subtract the whole F(a)F(a).
Substitution: change the limits
23× asked, last 2026
∫abf(g(x)) g′(x) dx=∫g(a)g(b)f(t) dt\int_a^bf(g(x))\,g^{\prime}(x)\,dx=\int_{g(a)}^{g(b)}f(t)\,dt
Never return to xx. Keep the dtdt factor: t=e2xt=e^{2x} gives dt=2t dxdt=2t\,dx.
Choosing the substitution
a+bsin⁡x+ccos⁡x: t=tan⁡x2;1−x2: x=sin⁡θa+b\sin x+c\cos x:\ t=\tan\frac x2;\qquad \sqrt{1-x^2}:\ x=\sin\theta
Bottom with every term of even degree nn in sin⁡x\sin x, cos⁡x\cos x: divide by cos⁡nx\cos^nx, t=tan⁡xt=\tan x.
Parts with limits
19× asked, last 2026
∫abuv′ dx=[uv]ab−∫abu′v dx∫abex(f+f′) dx=[exf]ab\begin{aligned}&\int_a^buv^{\prime}\,dx=[uv]_a^b-\int_a^bu^{\prime}v\,dx\\&\int_a^be^x(f+f^{\prime})\,dx=[e^xf]_a^b\end{aligned}
Minus before the second term: ∫0π/2xcos⁡x dx=π/2−1\int_0^{\pi/2}x\cos x\,dx=\pi/2-1. Spot exact derivatives: (fu)′(fu)^{\prime}, (P2)′=2PP′(P^2)^{\prime}=2PP^{\prime}.

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Practise Definite Integration

25 more formulas in 5 topics

Properties of definite integrals; Modulus, greatest integer, max and min; Leibniz rule and functions defined by integrals; Reduction formulas, Beta and estimation; and Integral as a limit of a sum. Read the first part free. Sign in free with Google to read the rest and download the PDF.

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Properties of definite integrals

  • Basic properties3×
  • King's rule24×
  • Pairs adding to a constant
  • Standard log integrals
  • Removing a stray x13×
  • Folding [0, 2a] at a8×
  • Even and odd functions15×
  • Denominator 1 + kᵍ, g odd17×
  • Periodic integrands15×
  • Inverse functions4×

Modulus, greatest integer, max and min

  • Modulus18×
  • [x] and {x}18×
  • [g(x)]21×
  • max and min5×

Leibniz rule and functions defined by integrals

  • Leibniz rule8×
  • x inside the integrand5×
  • Integral equation: variable limit11×
  • Integral equation: constant limits5×
  • Extrema of ∫ f4×
  • Limits by L'Hospital1×

Reduction formulas, Beta and estimation

  • Reduction formulas8×
  • Beta integral3×
  • Estimation and bounds14×

Integral as a limit of a sum

  • Riemann sum to integral15×
  • Limits of products

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About 225 past JEE Main questions from this chapter, with instant answer checking and step-by-step solutions where available. Free, and you can start without signing up.

Practise Definite Integration

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