Indefinite Integration — Class 12 Maharashtra HSC Mathematics revision notes
A 5-page visual revision sheet for Indefinite Integration, written for Class 12 Maharashtra HSC 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
Indefinite Integration in one look
differentiation run backward: a table of standard antiderivatives, then substitution, parts and partial fractions to reach it
Integration undoes Differentiation
- if d/dx[g(x)]=f(x), then ∫ f(x) dx = g(x)+c
- c is the CONSTANT OF INTEGRATION: a whole family of antiderivatives differs only by a vertical shift
- sums integrate term by term, and a constant pulls straight out: ∫ k f(x) dx = k∫ f(x) dx
Elementary Integration Formulae
- power rule, n not -1
∫ x^n dx = x^n+1/n+1+c - power rule with a linear inner function
∫(ax+b)^n dx = (ax+b)^n+1/n+1·1/a+c - reciprocal
∫1/ax+b dx = 1/alog|ax+b|+c - exponential
∫ e^ax+b dx = 1/ae^ax+b+c - sine
∫sin(ax+b) dx = -1/acos(ax+b)+c - cosine
∫cos(ax+b) dx = 1/asin(ax+b)+c - sec squared
∫sec^2(ax+b) dx = 1/atan(ax+b)+c - cosec squared
intcosec^2(ax+b) dx = -1/acot(ax+b)+c
Special cases of the power rule
- n=-1 — the power rule divides by zero, use log instead
- n=0 — the integrand is just 1
- a=1, b=0 in the (ax+b)^n family — it collapses to the plain power rule
- integrating a plain constant k — the result is a straight line in x
Power rule with a linear inner function
Q. Evaluate ∫(2x+3)^5 dx.
- n=5, a=2: apply ∫(ax+b)^n dx = (ax+b)^n+1/n+1·1/a
- = (2x+3)^6/6·1/2
Answer: (2x+3)^6/12 + c
Page 1 of 5. The full sheet, all 5 pages, opens free in the app.
What this Indefinite Integration sheet covers
- Integration undoes Differentiation
- Elementary Integration Formulae
- power rule, n not -1
- power rule with a linear inner function
- reciprocal
- exponential
- sine
- cosine
- sec squared
- cosec squared
- Special cases of the power rule
- Integration by Substitution
- if u=g(x) is differentiable and f is continuous on its range
- Substitution where the derivative is already present
- Integrals of Trigonometric Functions
- tan
- cot
- sec
- cosec
- Some Special Integrals
- 1
- 2
- 3
- 4
- 5
- 6
- 7
- Matching a denominator to the special-integral table
- A square-root denominator
- Integrals of the form: square root of (a squared plus or minus x squared)
- circle-arc form
- hyperbola-branch form
- same idea, sign flipped under the root
- Integration by Parts
- By parts with an algebraic and an exponential factor
- By parts with an algebraic and a trig factor
- Substitution vs integration by parts
- Substitution
- By parts
- The e^x[f(x)+f'(x)] Shortcut
- a function plus its own derivative, both riding on e^x
- Spotting the f plus f' pattern
- Integration by Partial Fractions
- Partial fractions with distinct linear factors
- Which technique to reach for
- A function beside its own derivative
- A product of an algebraic and a transcendental factor
- e^x multiplying a function plus its derivative
- A proper rational function with a factorable denominator
- 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 Indefinite Integration 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.