Indefinite Integral & Properties
∫ f(x) dx = F(x) + C ⇔ F'(x) = f(x)
- ∫ [af(x) + bg(x)] dx = a ∫ f(x) dx + b ∫ g(x) dx
- ∫ f(ax + b) dx = (1/a) F(ax + b) + C (a ≠ 0)
Standard Integrals (Must Memorise)
Power Formula
∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1)
Exponential
∫ eˣ dx = eˣ + C
∫ aˣ dx = aˣ/ln a + C
∫ aˣ dx = aˣ/ln a + C
Logarithmic
∫ dx/x = ln|x| + C
Trigonometric
∫ sin x dx = −cos x + C
∫ cos x dx = sin x + C
∫ tan x dx = −ln|cos x| + C
∫ sec²x dx = tan x + C
∫ cosec²x dx = −cot x + C
∫ sec x tan x dx = sec x + C
∫ cosec x cot x dx = −cosec x + C
∫ cos x dx = sin x + C
∫ tan x dx = −ln|cos x| + C
∫ sec²x dx = tan x + C
∫ cosec²x dx = −cot x + C
∫ sec x tan x dx = sec x + C
∫ cosec x cot x dx = −cosec x + C
Inverse Trig
∫ dx/√(a²−x²) = sin⁻¹(x/a + C
∫ dx/(a²+x²) = (1/a) tan⁻¹(x/a) + C
∫ dx/(x√(x²−a²)) = (1/a) sec⁻¹(x/a) + C
∫ dx/(a²+x²) = (1/a) tan⁻¹(x/a) + C
∫ dx/(x√(x²−a²)) = (1/a) sec⁻¹(x/a) + C
Methods of Integration
Substitution
Let x = g(t) → dx = g'(t) dt
By Parts
∫ u dv = u v − ∫ v du
Partial Fractions
For rational functions
Trig Substitutions
√(a²−x²) → x = a sinθ
√(a²+x²) → x = a tanθ
√(x²−a²) → x = a secθ
Definite Integral
∫_a^b f(x) dx = F(b) − F(a)
Newton-Leibniz Formula
If F'(x) = f(x) → ∫_a^b f(x) dx = [F(x)]_a^b = F(b) − F(a)
Properties of Definite Integrals
- ∫_a^b f(x) dx = ∫_a^b f(t) dt
- ∫_a^b f(x) dx = − ∫_b^a f(x) dx
- ∫_a^a f(x) dx = 0
- ∫_a^b f(x) dx = ∫_a^c f(x) dx + ∫_c^b f(x) dx
- ∫_a^b f(x) dx = ∫_a^b f(a+b−x) dx
- ∫_0^{2a} f(x) dx = ∫_0^a [f(x) + f(2a−x)] dx
- If f(2a−x) = f(x) → ∫_0^{2a} f(x) dx = 2 ∫_0^a f(x) dx
- If f(2a−x) = −f(x) → ∫_0^{2a} f(x) dx = 0
Area Under Curve
Area = ∫_a^b f(x) dx (f(x) ≥ 0)
Area between curves = ∫_a^b [f(x) − g(x)] dx
Area between curves = ∫_a^b [f(x) − g(x)] dx
JEE Main Tips & Weightage
One of the highest scoring chapters — 6–8 questions every year!
Most Important Topics
- By parts (ILATE rule)
- Definite integrals using properties (especially ∫_0^{2a} and f(a+b−x))
- Substitution & trig identities
- ∫ dx/(ax² + bx + c) forms
- Reduction formulae
- Area between curves & parametric forms
- Integration of irrational functions