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Cheat sheet · Class 12 · CBSE · Commerce Mathematics

Integrals cheat sheet

The whole chapter on one page: key formulas, facts and the points to remember, for quick revision before a test.

CBSE Class 12 · Mathematics

Cheat sheet

Chapter 7Integrals

ILATE for parts: Inverse, Log, Algebraic, Trig, Exponential.

1. Standard Integrals

∫xndx=xn+1n+1+C\int x^n dx = \dfrac{x^{n + 1}}{n + 1} + C∫1xdx=log⁡∣x∣+C\int\dfrac1x dx = \log|x| + C
∫dxx2+a2=1atan⁡−1xa+C\int\dfrac{dx}{x^2 + a^2} = \dfrac1a\tan^{-1}\dfrac xa + C∫dxa2−x2=sin⁡−1xa+C\int\dfrac{dx}{\sqrt{a^2 - x^2}} = \sin^{-1}\dfrac xa + C
∫dxx2−a2=12alog⁡∣x−ax+a∣+C\int\dfrac{dx}{x^2 - a^2} = \dfrac1{2a}\log\left|\dfrac{x - a}{x + a}\right| + C
∫dxx2±a2=log⁡∣x+x2±a2∣+C\int\dfrac{dx}{\sqrt{x^2 \pm a^2}} = \log\left|x + \sqrt{x^2 \pm a^2}\right| + C
∫a2−x2dx=x2a2−x2+a22sin⁡−1xa+C\int\sqrt{a^2 - x^2}dx = \dfrac x2\sqrt{a^2 - x^2} + \dfrac{a^2}2\sin^{-1}\dfrac xa + C

2. Methods

  • Substitution.
  • Partial fractions for rational functions.
  • By parts: ∫uv dx=u∫v dx−∫(u′∫v dx)dx\int uv\,dx = u\int v\,dx - \int\left(u'\int v\,dx\right)dx.
  • ∫ex[f(x)+f′(x)]dx=exf(x)+C\int e^x[f(x) + f'(x)]dx = e^xf(x) + C.

3. Definite Integrals

∫abf(x)dx=F(b)−F(a)\int_a^bf(x)dx = F(b) - F(a)
∫0af(x)dx=∫0af(a−x)dx\int_0^af(x)dx = \int_0^af(a - x)dx
∫−aaf=2∫0af\int_{-a}^{a}f = 2\int_0^af (even), 0 (odd)

4. Key Points

  • Don't forget + C.
  • Change limits when substituting.
  • Use properties to simplify definite integrals.
  • Integration is the reverse of differentiation.

Remember: Standard forms | substitution, partial fractions, parts | properties

Integrals: questions and notes

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