Maharashtra State Board · Class 12 · Mathematics · Part II Chapter 4

Definite Integration — Formula Sheet

Board Formulas

15 formulas

1.Definite Integral as Limit of a Sum★
$\int_{a}^{b} f (x) d x = lim_{n \to \infty} \sum_{r = 1}^{n} h f (a + r h), h = \frac{b - a}{n}$
$a, b$ : Lower and upper limits of integration · $n$ : Number of equal sub-intervals · $h$ : Width of each sub-interval, (b − a)/n
Divide [a, b] into n equal strips of width h, add the strip areas h·f(a + rh), then let n → ∞. Simplify the sum with the standard Σ results before taking the limit.
2.Standard Sums for Limit-of-Sum Questions
$\sum_{r = 1}^{n} r = \frac{n ( n + 1 )}{2}, \sum_{r = 1}^{n} r^{2} = \frac{n ( n + 1 ) ( 2 n + 1 )}{6}, \sum_{r = 1}^{n} r^{3} = [\frac{n ( n + 1 )}{2}]^{2}$
For an exponential integrand the sum is a G.P.: use a(rⁿ − 1)/(r − 1) and the limit (eʰ − 1)/h → 1 as h → 0.
3.Fundamental Theorem of Integral Calculus★
$\int_{a}^{b} f (x) d x = [F (x)]_{a}^{b} = F (b) - F (a), F^{'} (x) = f (x)$
$F (x)$ : Any antiderivative (indefinite integral) of f(x)
Find any antiderivative F, substitute the upper limit, then subtract the value at the lower limit. No constant of integration is needed — it cancels.
4.Property: Change of Variable Name
$\int_{a}^{b} f (x) d x = \int_{a}^{b} f (t) d t$
The variable of integration is a dummy variable. This is what lets you rename t back to x at the end of every property proof.
5.Property: Interchanging the Limits
$\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, \int_{a}^{a} f (x) d x = 0$
Swapping the limits changes the sign. Used in almost every proof after a substitution reverses the limits.
6.Property: Splitting the Interval
$\int_{a}^{b} f (x) d x = \int_{a}^{c} f (x) d x + \int_{c}^{b} f (x) d x, a < c < b$
Use it for modulus functions, greatest-integer functions and piecewise definitions — split at the point where the formula of f changes.
7.Property: a + b − x★
$\int_{a}^{b} f (x) d x = \int_{a}^{b} f (a + b - x) d x$
Works for any limits a to b. Typical use: an integrand like √x/(√x + √(a + b − x)), where the two integrands add up to 1, so 2I = b − a.
8.Property: a − x (lower limit zero)★
$\int_{0}^{a} f (x) d x = \int_{0}^{a} f (a - x) d x$
The special case of the a + b − x property with lower limit 0. With a = π/2 it swaps sin x and cos x.
9.Property: Splitting 0 to 2a
$\int_{0}^{2 a} f (x) d x = \int_{0}^{a} f (x) d x + \int_{0}^{a} f (2 a - x) d x$
Split at a, then substitute x = 2a − t in the second piece. Leads directly to the next property.
10.Property: 0 to 2a with f(2a − x) = ± f(x)
$\int_{0}^{2 a} f (x) d x = {2 \int_{0}^{a} f (x) d x, 0, f (2 a - x) = f (x) f (2 a - x) = - f (x)$
Example: sin(π − x) = sin x, so ∫₀^π sin x dx = 2∫₀^{π/2} sin x dx = 2. And cos(π − x) = −cos x, so ∫₀^π cos x dx = 0.
11.Property: Even and Odd Functions★
$\int_{- a}^{a} f (x) d x = {2 \int_{0}^{a} f (x) d x, 0, f (- x) = f (x) (even) f (- x) = - f (x) (odd)$
Test f(−x) for the whole integrand. x², cos x, |x| are even; x³, sin x, tan x are odd; odd × even is odd, odd × odd is even.
12.Substitution in a Definite Integral
$\int_{a}^{b} f (g (x)) g^{'} (x) d x = \int_{g (a)}^{g (b)} f (u) d u, u = g (x)$
Change the limits to u = g(a) and u = g(b) as soon as you substitute; then you never need to return to x.
13.Integration by Parts for Definite Integrals
$\int_{a}^{b} u v d x = [u \int v d x]_{a}^{b} - \int_{a}^{b} [\frac{d u}{d x} \int v d x] d x$
Choose u by the ILATE order, as in indefinite integration. Apply the limits to the first term as well as to the remaining integral.
14.Standard Result: sinⁿ x over sinⁿ x + cosⁿ x
$\int_{0}^{π /2} \frac{s i n ^{n} x}{s i n ^{n} x + c o s ^{n} x} d x = \int_{0}^{π /2} \frac{c o s ^{n} x}{s i n ^{n} x + c o s ^{n} x} d x = \frac{π}{4}$
Holds for any power n (including n = 1/2, i.e. √sin x). The same idea works with tan x and cot x.
15.Standard Result: ∫ log sin x from 0 to π/2
$\int_{0}^{π /2} lo g sin x d x = \int_{0}^{π /2} lo g cos x d x = - \frac{π}{2} lo g 2$
Here log means natural logarithm (base e), as in the textbook. Learn the proof — it uses three different properties.