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Indefinite Integration revision notes

Every idea in the Indefinite 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 22 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.

Standard integrals and basic rules

Antiderivative and the constant C
16× asked, last 2026
∫f(x) dx=F(x)+C  ⟺  F′(x)=f(x),∫(af+bg) dx=a∫f dx+b∫g dx\int f(x)\,dx=F(x)+C \iff F'(x)=f(x),\qquad \int (af+bg)\,dx=a\int f\,dx+b\int g\,dx
One CC on each interval of the domain. Fix it from a given value or limit of the full antiderivative.
xxyy2ln⁡x+C1\ln x+C_1ln⁡(−x)+C2\ln(-x)+C_2all slope 12\frac12 at x=2x=2
∫dxx=ln⁡∣x∣\int\frac{dx}x=\ln|x|: shifting by CC keeps every slope; each side of 0 gets its own CC
  • Not substituting into a form that is undefined at the given point. −6/(1−cot⁡x)-6/(1-\cot x) has no value at x=0x=0. Rewrite it as 6tan⁡x/(1−tan⁡x)6\tan x/(1-\tan x) (equal wherever both exist), then use I(0)I(0).
  • Not stopping at CC. After f(0)f(0) fixes CC, add it to the other terms at the new point; in f(1)=a+bln⁡3f(1)=a+b\ln3 the number aa already contains CC. A limit like lim⁡x→∞I(x)=0\lim_{x\to\infty}I(x)=0 fixes CC the same way.
  • Quick method. Asked g′(a)g'(a) where g=∫f dxg=\int f\,dx? It is just f(a)f(a), no integration needed. Asked g(b)−g(a)g(b)-g(a), with a,ba,b in one interval of the domain? CC cancels, so no given value is needed.
Power, exponential and log integrals
2× asked, last 2025
∫xn dx=xn+1n+1 (n≠−1),∫dxx=ln⁡∣x∣,∫ax dx=axln⁡a\int x^n\,dx=\frac{x^{n+1}}{n+1}\ (n\ne-1),\qquad \int\frac{dx}{x}=\ln|x|,\qquad \int a^x\,dx=\frac{a^x}{\ln a}
ln⁡∣x∣\ln|x| on intervals without 0; a>0a>0, a≠1a\ne1; ∫ex dx=ex\int e^x\,dx=e^x. If ∫f=F\int f=F, then ∫f(ax+b) dx=F(ax+b)/a\int f(ax+b)\,dx=F(ax+b)/a.
  • Not x3x^3 for e3ln⁡2xe^{3\ln 2x}. enln⁡u=une^{n\ln u}=u^n keeps all of uu, so it is (2x)3=8x3(2x)^3=8x^3. Rewrite such terms as powers before you integrate.
Trigonometric standard integrals
2× asked, last 2022
∫tan⁡x dx=ln⁡∣sec⁡x∣,∫sec⁡x dx=ln⁡∣sec⁡x+tan⁡x∣,∫csc⁡x dx=ln⁡∣tan⁡x2∣\int\tan x\,dx=\ln|\sec x|,\qquad \int\sec x\,dx=\ln|\sec x+\tan x|,\qquad \int\csc x\,dx=\ln\left|\tan\frac x2\right|
Also ∫cot⁡x=ln⁡∣sin⁡x∣\int\cot x=\ln|\sin x|, ∫sec⁡2x=tan⁡x\int\sec^2x=\tan x, ∫csc⁡2x=−cot⁡x\int\csc^2x=-\cot x, ∫sec⁡xtan⁡x=sec⁡x\int\sec x\tan x=\sec x.
xxyyπ2\frac\pi2π\pi1ln⁡∣tan⁡x2∣\ln|\tan\frac x2| risesln⁡∣cot⁡x2∣\ln|\cot\frac x2| fallscosec⁡x>0\operatorname{cosec}x>0
On (0,π)(0,\pi) the integrand cosec⁡x\operatorname{cosec}x is positive, so the right answer must rise
  • Not ln⁡∣cot⁡(x/2)∣\ln|\cot(x/2)| for ∫csc⁡x dx\int\csc x\,dx. It is ln⁡∣tan⁡(x/2)∣=ln⁡∣csc⁡x−cot⁡x∣\ln|\tan(x/2)|=\ln|\csc x-\cot x|. Check: on (0,π)(0,\pi) the integrand is positive, so the answer must increase, and tan⁡(x/2)\tan(x/2) does.
Matching a given answer form
8× asked, last 2025
ddx(given answer)=integrand ⇒ compare coefficients\frac{d}{dx}\big(\text{given answer}\big)=\text{integrand}\ \Rightarrow\ \text{compare coefficients}
For answers with unknown constants. Differentiating the form is quicker and safer than integrating from scratch.
  • Not the first constant you find. Answer what is asked (B/AB/A, ABAB, α+γ/β\alpha+\gamma/\beta). Equal antiderivatives can look different: flipping the fraction inside a log flips the sign of its coefficient.
  • Quick method. Clear the root before comparing. If ∫(2x2+5x+9) dx/Q=(x+α)Q+βln⁡∣x+1/2+Q∣\int(2x^2+5x+9)\,dx/\sqrt Q=(x+\alpha)\sqrt Q+\beta\ln|x+1/2+\sqrt Q|, Q=x2+x+1Q=x^2+x+1, then 2Q×2\sqrt Q\times derivative =4x2+(3+2α)x+2+α+2β=4x2+10x+18=4x^2+(3+2\alpha)x+2+\alpha+2\beta=4x^2+10x+18.

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Practise Indefinite Integration

18 more concepts in 5 topics

Special integrals and quadratics, Integration by substitution, Trigonometric integrals, Integration by parts and Partial fractions. Read the first part free. Sign in free with Google to read the rest and download the PDF.

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Special integrals and quadratics

  • 1/(x² ± a²) and 1/(a² − x²)7×
  • 1/√(a² − x²) and 1/√(x² ± a²)3×
  • √(x² ± a²) and √(a² − x²)1×
  • Completing the square5×
  • Numerator split: A·(derivative) + B4×

Integration by substitution

  • Derivative on top: f′/f and fⁿ·f′10×
  • Taking out a power: xⁿ and (x − a)ᵐ(x − b)ⁿ8×
  • Trig and root substitutions7×
  • x ± 1/x substitution3×

Trigonometric integrals

  • Powers and products of sin and cos4×
  • Divide by cos²x; t = tan(x/2)4×
  • sin x ± cos x with sin 2x3×

Integration by parts

  • Integration by parts (ILATE)4×
  • eˣ(f + f′) and its general form7×
  • Hidden product rule: f + xf′3×
  • Reduction: powers of sec, cosec, tan, cot1×

Partial fractions

  • Partial fractions: the three forms6×
  • Improper fractions and the x² = t trick3×

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