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Indefinite Integration formula sheet

Every formula in the Indefinite Integration chapter for JEE Main, with one line on when it applies. Topic 1 is open to everyone; sign in free to read all 26 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.

Standard integrals and basic rules

Antiderivative and C
16× asked, last 2026
∫f dx=F+C  ⟺  F′=f∫(af+bg) dx=a∫f dx+b∫g dx\begin{aligned}&\int f\,dx=F+C\iff F^{\prime}=f\\&\int(af+bg)\,dx=a\int f\,dx+b\int g\,dx\end{aligned}
One CC per interval of the domain; fix it from a given value or limit. g=∫fg=\int f: g′(a)=f(a)g^{\prime}(a)=f(a), and g(b)−g(a)g(b)-g(a) needs no CC.
Power, exponential and log
2× asked, last 2025
∫xn dx=xn+1n+1 (n≠−1),∫dxx=ln⁡∣x∣∫ax dx=axln⁡a\begin{aligned}&\int x^n\,dx=\frac{x^{n+1}}{n+1}\ (n\ne-1),\qquad \int\frac{dx}x=\ln|x|\\&\int a^x\,dx=\frac{a^x}{\ln a}\end{aligned}
∫f(ax+b) dx=F(ax+b)/a\int f(ax+b)\,dx=F(ax+b)/a. Rewrite as powers first: e3ln⁡2x=(2x)3e^{3\ln2x}=(2x)^3.
Trigonometric
2× asked, last 2022
∫tan⁡x dx=ln⁡∣sec⁡x∣,∫cot⁡x dx=ln⁡∣sin⁡x∣∫sec⁡x dx=ln⁡∣sec⁡x+tan⁡x∣∫cosec⁡x dx=ln⁡∣tan⁡x2∣=ln⁡∣cosec⁡x−cot⁡x∣\begin{aligned}&\int\tan x\,dx=\ln|\sec x|,\qquad \int\cot x\,dx=\ln|\sin x|\\&\int\sec x\,dx=\ln|\sec x+\tan x|\\&\int\operatorname{cosec}x\,dx=\ln\left|\tan\frac x2\right|=\ln|\operatorname{cosec}x-\cot x|\end{aligned}
∫sec⁡2x=tan⁡x\int\sec^2x=\tan x, ∫cosec⁡2x=−cot⁡x\int\operatorname{cosec}^2x=-\cot x, ∫sec⁡xtan⁡x=sec⁡x\int\sec x\tan x=\sec x, ∫cosec⁡xcot⁡x=−cosec⁡x\int\operatorname{cosec}x\cot x=-\operatorname{cosec}x.
Matching a given answer
8× asked, last 2025
ddx(given answer)=integrand⇒compare coefficients\begin{aligned}&\frac d{dx}(\text{given answer})=\text{integrand}\\&\Rightarrow\text{compare coefficients}\end{aligned}
Answer what is asked (B/AB/A, ABAB). Clear roots before comparing. Flipping a log’s fraction flips its coefficient’s sign.

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Practise Indefinite Integration

22 more formulas 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), 1/(a² − x²)7×
  • 1/√(a² − x²), 1/√(x² ± a²)3×
  • √(x² ± a²), √(a² − x²)1×
  • Completing the square5×
  • Numerator split4×

Integration by substitution

  • Derivative on top10×
  • Taking out a power8×
  • xᵏ(1 + xⁿ)ᵖ below
  • Trig and root substitutions7×
  • x ± 1/x substitution3×

Trigonometric integrals

  • Powers and products4×
  • Divide by cos²x4×
  • t = tan(x/2)
  • sin x ± cos x with sin 2x3×

Integration by parts

  • Integration by parts4×
  • eᵃˣ sin bx, eᵃˣ cos bx
  • eˣ(f + f′)7×
  • Hidden product rule3×
  • Reduction: cosecⁿ, secⁿ1×
  • Reduction: tanⁿ, cotⁿ

Partial fractions

  • The three forms6×
  • Improper fractions3×

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About 60 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 Indefinite Integration

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