CBSE · Class 12 · Mathematics · Chapter 7

Integrals — Formula Sheet

Board Formulas

14 formulas

1.Power Rule★
$\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + C, n \neq = - 1$
Raise power by one, divide by new power. Fails at n = −1 (that case gives ln|x|).
2.Integral of 1/x
$\int \frac{1}{x} d x = ln ∣ x ∣ + C$
The missing case of the power rule. Absolute value is essential — x can be negative.
3.Exponential and Logarithm
$\int e^{x} d x = e^{x} + C, \int a^{x} d x = \frac{a ^{x}}{l n a} + C$
e^x is invariant. General a^x picks up 1/ln a in the antiderivative.
4.Standard Trig Integrals★
$\int sin x d x = - cos x + C, \int cos x d x = sin x + C, \int sec^{2} x d x = tan x + C$
sin flips sign, cos does not. sec² integrates cleanly to tan.
5.Inverse Trig Integrals★
$\int \frac{d x}{1 - x ^{2}} = sin^{- 1} x + C, \int \frac{d x}{1 + x ^{2}} = tan^{- 1} x + C$
Recognise the RHS by shape of the integrand. Generalisation: replace 1 with a² and divide by a where appropriate.
6.Integration by Substitution★
$\int f (g (x)) g^{'} (x) d x = \int f (u) d u, u = g (x)$
Reverses the chain rule. Look for a function whose derivative is also present (up to a constant).
7.Integration by Parts★
$\int u d v = uv - \int v d u$
ILATE rule chooses u: Inverse trig, Logarithm, Algebraic, Trig, Exponential (earlier letter → u).
8.Partial Fractions (Linear factors)★
$\frac{P ( x )}{( x - a ) ( x - b )} = \frac{A}{x - a} + \frac{B}{x - b}$
Cover-up method: A = P(a)/(a−b), B = P(b)/(b−a). Works when degree(P) < degree(denominator).
9.Standard Form: 1/(x² + a²)★
$\int \frac{d x}{x ^{2} + a ^{2}} = \frac{1}{a} tan^{- 1} (\frac{x}{a}) + C$
Rule of thumb: sum in denominator ⇒ arctan.
10.Standard Form: 1/(x² − a²)
$\int \frac{d x}{x ^{2} - a ^{2}} = \frac{1}{2 a} ln \frac{x - a}{x + a} + C$
Difference in denominator ⇒ log of ratio.
11.Standard Form: 1/√(a² − x²)
$\int \frac{d x}{a ^{2} - x ^{2}} = sin^{- 1} (\frac{x}{a}) + C$
Square-root difference ⇒ arcsin.
12.Definite Integral (Fundamental Theorem)★
$\int_{a}^{b} f (x) d x = F (b) - F (a), F^{'} (x) = f (x)$
Net signed area under y = f(x) from x = a to x = b.
13.Property: Reflection★
$\int_{0}^{a} f (x) d x = \int_{0}^{a} f (a - x) d x$
Substitute t = a − x. Extremely powerful for symmetric integrands like x/(x + (a − x)).
14.Property: Even/Odd
$\int_{- a}^{a} f (x) d x = {2 \int_{0}^{a} f (x) d x 0 f even f odd$
Odd function on symmetric limits gives 0 — no work needed. Check parity first.