Integrals — Class 12 CBSE Mathematics revision notes
A 3-page visual revision sheet for Integrals, written for Class 12 CBSE Mathematics. Formulas, diagrams and the traps that cost marks, laid out to be read the night before rather than the term before. Free, and it opens inside CapCompass.
Page 1 as PDF
Integrals in one look
differentiation undone, one antiderivative at a time
Integration undoes differentiation
- if F is an antiderivative of f
∫ f(x) dx = F(x) + C - standard power rule, for n not equal to -1
∫ x^n dx = x^n+1/n+1 + C - the one exponent the power rule skips
∫ 1/x dx = ln|x| + C
More standard integrals
- exponential
∫ e^x dx = e^x + C - the two inverse-trig integrals that show up constantly
∫ 1/1+x^2 dx = tan^-1x + C - and the other one
∫ frac1√1-x^2 dx = sin^-1x + C
The constant of integration
- an indefinite integral is not one function, it's the whole family sharing the same derivative
- two antiderivatives of f differ only by a constant, since a constant's derivative is 0
- dropping +C on a definite integral costs nothing; an indefinite one loses its whole family
Indefinite vs definite integral
- a whole family of functions
- always carries +C
- a single number, F(b)-F(a)
- no +C, the limits fix it
Integration by substitution
- pick a part of the integrand to call u
u = g(x) - rewrite dx in terms of du
du = g'(x) dx - the integral turns into one purely in u
∫ f(g(x)) g'(x) dx = ∫ f(u) du
Integrate by substitution
Q. Evaluate the integral of 2x(x^2+1)^3 with respect to x.
- let u = x^2+1, so du = 2x dx
- the integrand 2x(x^2+1)^3 dx becomes u^3 du exactly
- integrate: the integral of u^3 du is u^4/4 + C
- substitute back u = x^2+1
Answer: (x^2+1)^4 / 4 + C
Page 1 of 3. The full sheet, all 3 pages, opens free in the app.
What this Integrals sheet covers
- Integration undoes differentiation
- if F is an antiderivative of f
- standard power rule, for n not equal to -1
- the one exponent the power rule skips
- More standard integrals
- exponential
- the two inverse-trig integrals that show up constantly
- and the other one
- The constant of integration
- Indefinite vs definite integral
- Indefinite, no limits
- Definite, with limits
- Integration by substitution
- pick a part of the integrand to call u
- rewrite dx in terms of du
- the integral turns into one purely in u
- Integrate by substitution
- Integration by parts
- for two functions u and v of x
- picking which factor is u: inverse trig, log, algebraic, trig, exponential
- Integrate by parts
- Partial fractions for rational integrands
- a proper rational function with distinct linear factors on the bottom
- a repeated linear factor needs one extra term per repetition
- Which integration technique fits
- Definite integrals
- the fundamental theorem: F is any antiderivative of f
- Evaluate a definite integral
- Properties of definite integrals
- reversing the limits flips the sign
- equal limits give zero, always
- splitting the interval at any point c between a and b
- the reflection trick, reading the interval backward
- The symmetric-interval special cases
- Reading the chapter together
- Traps that cost marks
- Tricks that save time
How CapCompass uses this sheet
These notes are not a standalone PDF. CapCompass plans Mathematics chapter by chapter against your real exam dates and free hours, and when the plan reaches Integrals the sheet opens inside the study block itself — at the page covering the sub-topic that block is on, with the timer still running. Revision arrives at the moment the plan says revise, instead of having to be gone looking for.