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Definite Integration revision notes

Every idea in the Definite Integration chapter for JEE Main, with the usual mistakes and the fast methods. Topic 1 is open to everyone; sign in free to read all 25 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 concept, and "last" is the latest year one appeared. ★ marks ideas asked 8 or more times.

Fundamental theorem and evaluation

Fundamental theorem of calculus
20× asked, last 2026
∫abf(x) dx=F(b)−F(a),F′(x)=f(x) on [a,b]\int_a^b f(x)\,dx = F(b)-F(a),\qquad F'(x)=f(x)\ \text{on}\ [a,b]
ff continuous on [a,b][a,b] and F′=fF'=f on all of [a,b][a,b], so FF has no jump inside. The constant CC cancels.
xxyy−1-11tan⁡−1x\tan^{-1}x: π2\frac\pi2−tan⁡−11x-\tan^{-1}\frac1x jumps by π\pi: gives −π2-\frac\pi2
Same derivative 11+x2\frac1{1+x^2}, but only tan⁡−1x\tan^{-1}x has no jump in [−1,1][-1,1]: ∫−11=π2\int_{-1}^1=\frac\pi2
  • Not F(b)−F(a)F(b)-F(a) with an antiderivative that jumps inside [a,b][a,b]. −tan⁡−1(1/x)-\tan^{-1}(1/x) has derivative 1/(1+x2)1/(1+x^2) but jumps at 0, giving −π/2-\pi/2 on [−1,1][-1,1]; the true value is π/2\pi/2.
  • Not F(b)F(b) alone or F(b)+F(a)F(b)+F(a). Subtract the whole value at the lower limit, every factor included: [exxx]12=4e2−e[e^xx^x]_1^2=4e^2-e, not 4e2−14e^2-1.
Substitution: change the limits
23× asked, last 2026
∫abf(g(x)) g′(x) dx=∫g(a)g(b)f(t) dt\int_a^b f\big(g(x)\big)\,g'(x)\,dx=\int_{g(a)}^{g(b)} f(t)\,dt
g′g' continuous on [a,b][a,b]. Change the limits along with the variable; never return to xx.
  • Not dropping the factor from dtdt. t=e2xt=e^{2x} gives dt=2t dxdt=2t\,dx, and t=2xt=2x gives dx=dt/2dx=dt/2. A lost factor doubles or halves the answer.
  • Quick method. Pick by shape: t=tan⁡(x/2)t=\tan(x/2) for a+bsin⁡x+ccos⁡xa+b\sin x+c\cos x; for 1/P1/P with every term of PP of the same even degree nn in sin⁡x\sin x, cos⁡x\cos x, divide by cos⁡nx\cos^nx and put t=tan⁡xt=\tan x; x=sin⁡θx=\sin\theta for 1−x2\sqrt{1-x^2}.
Integration by parts with limits
19× asked, last 2026
∫abu v′ dx=[u v]ab−∫abu′ v dx,∫abex(f+f′) dx=[exf]ab\int_a^b u\,v'\,dx=\big[u\,v\big]_a^b-\int_a^b u'\,v\,dx,\qquad \int_a^b e^x\big(f+f'\big)\,dx=\big[e^xf\big]_a^b
uu, vv differentiable on [a,b][a,b]. The boundary term uses both limits and often vanishes.
  • Not ++ before the second term. It is uv−∫u′vuv-\int u'v, and many wrong options are one sign slip: ∫0π/2xcos⁡x dx=π/2−1\int_0^{\pi/2}x\cos x\,dx=\pi/2-1, not π/2+1\pi/2+1.
  • Quick method. Spot an exact derivative first: f′u+fu′=(fu)′f'u+fu'=(fu)', 2PP′=(P2)′2PP'=(P^2)', tan⁡nxsec⁡2x=(tan⁡n+1x)′/(n+1)\tan^nx\sec^2x=(\tan^{n+1}x)'/(n+1). Then only end values remain.

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Practise Definite Integration

22 more concepts 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×
  • Removing a stray x13×
  • Folding [0, 2a] at a8×
  • Even and odd functions15×
  • Denominator 1 + k^g with g odd17×
  • Periodic integrands15×
  • Reciprocal limits and inverse functions4×

Modulus, greatest integer, max and min

  • Modulus: split where the sign changes18×
  • [x] and {x} over whole units18×
  • [g(x)]: cut where g(x) is an integer21×
  • max, min and piecewise definitions5×

Leibniz rule and functions defined by integrals

  • Leibniz rule8×
  • x inside the integrand5×
  • Integral equations: variable limit11×
  • Integral equations: constant limits5×
  • Extrema of F(x) = ∫ f(t) dt4×
  • Limits of integrals (L'Hospital)1×

Reduction formulas, Beta and estimation

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

Integral as a limit of a sum

  • Riemann sum to integral15×

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