Maharashtra State Board · Class 12 · Mathematics · Part II Chapter 3

Integration — Formula Sheet

Board Formulas

12 formulas

1.Fundamental Theorem of Calculus★
$\int_{a}^{b} f (x) d x = F (b) - F (a), F^{'} (x) = f (x)$
Links definite integration to antiderivatives. F is any antiderivative of f.
2.Standard Integrals★
$\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + C (n \neq = - 1); \int \frac{d x}{x} = ln ∣ x ∣ + C$
Basic table; add +C in indefinite form.
3.Trigonometric Integrals
$\int sin x d x = - cos x + C; \int cos x d x = sin x + C; \int sec^{2} x d x = tan x + C$
Memorise all six. Sign errors are the most common HSC mistake.
4.Exponential and Log Integrals
$\int e^{x} d x = e^{x} + C; \int a^{x} d x = \frac{a ^{x}}{l n a} + C$
Note the ln a in denominator for general base a.
5.Integration by Substitution★
$\int f (g (x)) g^{'} (x) d x = \int f (u) d u with u = g (x)$
Change variable so that du = g'(x) dx and integrand becomes tractable.
6.Integration by Parts (ILATE)★
$\int u d v = uv - \int v d u$
ILATE order: Inverse trig, Logarithmic, Algebraic, Trigonometric, Exponential. Choose u higher up in the list.
7.Partial Fractions★
$\frac{P ( x )}{( x - a ) ( x - b )} = \frac{A}{x - a} + \frac{B}{x - b}$
Compute A, B by cover-up rule or comparing coefficients. For repeated roots include (x−a)² term as well.
8.Definite Integral Property (1)★
$\int_{a}^{b} f (x) d x = \int_{a}^{b} f (a + b - x) d x$
Powerful trick for symmetric integrands. Also (0,a): ∫f(x) = ∫f(a−x).
9.Definite Integral Property (2 — even/odd)
$\int_{- a}^{a} f (x) d x = {2 \int_{0}^{a} f (x) d x 0 f even f odd$
Recognise sin, cos, x symmetries. Saves half the work.
10.Reduction Formulae
$\int sin^{n} x d x = - \frac{s i n ^{n - 1} x c o s x}{n} + \frac{n - 1}{n} \int sin^{n - 2} x d x$
Repeated integration by parts. Similar formula for cosⁿ x.
11.Definite Integral as Limit of a Sum★
$\int_{a}^{b} f (x) d x = lim_{n \to \infty} \sum_{r = 1}^{n} f (a + r h) \cdot h, h = \frac{b - a}{n}$
Riemann sum definition. Use Σr = n(n+1)/2, Σr² = n(n+1)(2n+1)/6.
12.Common Trigonometric Substitutions
$a^{2} - x^{2} : x = a sin θ; a^{2} + x^{2} : x = a tan θ; x^{2} - a^{2} : x = a sec θ$
Choose based on the surd. Simplifies to a·cos, a·sec, a·tan respectively.