Concept Core
From basic integrals to integration by parts and definite integral properties.
Standard Integrals — Must Know
∫xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1) ∫1/x dx = ln|x| + C
∫eˣ dx = eˣ + C ∫aˣ dx = aˣ/ln a + C
∫sin x dx = −cos x + C ∫cos x dx = sin x + C
∫sec²x dx = tan x + C ∫1/√(1−x²) dx = sin⁻¹x + C
∫1/(1+x²) dx = tan⁻¹x + C ∫1/(a²+x²) dx = (1/a)tan⁻¹(x/a)+C
Substitution Method
When integrand has f(g(x))×g'(x): substitute u = g(x), du = g'(x)dx
Example: ∫2x e^(x²) dx. Let u = x², du = 2x dx. → ∫eᵘ du = eᵘ + C = e^(x²) + C
Standard substitution results:
∫f'(x)/f(x) dx = ln|f(x)| + C
∫[f(x)]ⁿ f'(x) dx = [f(x)]ⁿ⁺¹/(n+1) + C
Integration by Parts (IBP)
∫u dv = uv − ∫v du
Choose u using ILATE priority: Inverse trig → Logarithm → Algebraic → Trig → Exponential. The first type in ILATE is u, the rest is dv.
Example: ∫x eˣ dx → u=x, dv=eˣ dx → uv−∫v du = xeˣ − ∫eˣ dx = xeˣ − eˣ + C
Definite Integral Properties
∫ₐᵇ f(x)dx = −∫ᵦₐ f(x)dx
∫ₐᵇ f(x)dx = ∫ₐᶜ f(x)dx + ∫ᶜᵇ f(x)dx
∫ₐᵃ f(x)dx = 0
∫₀ᵃ f(x)dx = ∫₀ᵃ f(a−x)dx (King's Property)
King's Property is extremely useful for symmetric integrals and direct cancellation.
Area Under a Curve
Area between curve y=f(x) and x-axis from a to b:
A = ∫ₐᵇ |f(x)| dx
The absolute value ensures positive area when curve is below x-axis. Split the integral at zeros of f(x) where the sign changes.
Area between two curves: A = ∫ₐᵇ [f(x) − g(x)] dx (f above g).
Special Integrals by Reduction
∫√(a²−x²) dx = (x/2)√(a²−x²) + (a²/2)sin⁻¹(x/a) + C
∫1/√(x²+a²) dx = ln|x + √(x²+a²)| + C
∫eˣ[f(x)+f'(x)] dx = eˣ f(x) + C
The last one is particularly useful — when an eˣ factor multiplies f(x)+f'(x), the integral is simply eˣ f(x).
Formula Vault
All integration formulas for EAPCET.
Power Integral
∫xⁿ dx = xⁿ⁺¹/(n+1) + C
n ≠ −1; don't forget +C
Exponential
∫eˣ dx = eˣ + C
∫aˣ dx = aˣ/ln(a) + C
Reciprocal
∫1/x dx = ln|x| + C
Absolute value important
sin and cos
∫sin x dx = −cos x + C
∫cos x dx = sin x + C
sec² and cosec²
∫sec²x dx = tan x + C
∫cosec²x dx = −cot x + C
Inverse Trig
∫1/√(1−x²) dx = sin⁻¹x + C
∫1/(1+x²) dx = tan⁻¹x + C
Logarithmic Form
∫f'(x)/f(x) dx = ln|f(x)| + C
Numerator = derivative of denominator
Integration by Parts
∫u dv = uv − ∫v du
ILATE rule for choosing u
King's Property
∫₀ᵃ f(x)dx = ∫₀ᵃ f(a−x)dx
Extremely useful for definite integrals
eˣ Composite
∫eˣ[f(x)+f'(x)]dx = eˣf(x)+C
Recognise this pattern for instant answer
Worked Examples
5 problems — substitution, by parts, definite integral, King's property, and a trap.
EasyIntegrate ∫(3x² + 2x − 5)dx▾
Find ∫(3x² + 2x − 5) dx.
1
Apply power rule to each term:
2
= 3x³/3 + 2x²/2 − 5x + C = x³ + x² − 5x + C
✓ ∫ = x³ + x² − 5x + C
EasyFind ∫(2x/(x²+1))dx▾
Evaluate ∫2x/(x²+1) dx.
1
Numerator 2x is the derivative of denominator (x²+1).
2
Using ∫f'/f dx = ln|f| + C:
3
= ln|x²+1| + C = ln(x²+1) + C (x²+1 always positive)
✓ ∫ = ln(x²+1) + C
MediumIntegrate ∫x eˣ dx by parts▾
Evaluate ∫x eˣ dx using integration by parts.
1
ILATE: u = x (Algebraic), dv = eˣ dx (Exponential)
3
∫x eˣ dx = uv − ∫v du = xeˣ − ∫eˣ dx = xeˣ − eˣ + C = eˣ(x−1) + C
✓ ∫x eˣ dx = eˣ(x−1) + C
EAPCET LevelUse King's property: ∫₀^(π/2) sin x/(sin x + cos x) dx▾
Evaluate ∫₀^(π/2) sin x/(sin x + cos x) dx.
1
Let I = ∫₀^(π/2) sinx/(sinx+cosx) dx
2
By King's property: f(x) → f(π/2−x): sin(π/2−x)=cosx, cos(π/2−x)=sinx
3
I = ∫₀^(π/2) cosx/(cosx+sinx) dx
4
Add both integrals: 2I = ∫₀^(π/2) (sinx+cosx)/(sinx+cosx) dx = ∫₀^(π/2) 1 dx = π/2
✓ I = π/4 (King's property approach)
Trap Question∫₀^1 x/(1+x²) dx — students add constant C to definite integral▾
Evaluate the definite integral ∫₀¹ x/(1+x²) dx. ⚠️ Classic formatting trap.
1
The trap: Adding arbitrary constant C to a definite integral. Definite integrals have a specific numerical value — no C.
2
Let u = 1+x², du = 2x dx → x dx = du/2
3
= (1/2)∫₁² 1/u du = (1/2)[ln u]₁² = (1/2)(ln 2 − ln 1) = (1/2) ln 2
✓ ∫₀¹ x/(1+x²) dx = (ln 2)/2 — no +C for definite integrals
Mistake DNA
4 integration errors from EAPCET distractor analysis.
➕
Adding +C to Definite Integrals
Definite integrals give a specific numerical value. The constant of integration appears only in indefinite integrals.
❌ Wrong
∫₀¹ x dx = [x²/2 + C]₀¹
= (1/2+C) − (0+C) ✗
(C cancels anyway but
adding it is wrong form)
✓ Correct
∫₀¹ x dx = [x²/2]₀¹
= 1/2 − 0 = 1/2 ✓
No +C for definite ✓
Indefinite integral: f(x) + C (unknown constant; family of curves). Definite integral: specific number between limits. C is not needed for definite integrals.
🔢
Forgetting n+1 in Denominator for Power Rule
∫xⁿ dx = xⁿ⁺¹/(n+1) + C. Students raise the power but forget to divide by the new exponent.
❌ Wrong
∫x³ dx = x⁴ + C ✗
(forgot to divide by 4)
✓ Correct
∫x³ dx = x⁴/4 + C ✓
Increase power AND
divide by new power
Power rule for integration: raise the exponent by 1, then divide by the new exponent. These two steps always go together.
🌊
ILATE: Choosing Wrong Function as u in IBP
Choosing a poor u leads to a more complex integral. ILATE gives the optimal choice.
❌ Wrong
∫x sin x dx: u=sin x,
dv=x dx ✗
→ makes integral harder
✓ Correct
ILATE: A before T ✓
u=x (Algebraic)
dv=sinx dx ✓
→ simpler integral
ILATE order: Inverse trig, Logarithm, Algebraic, Trigonometric, Exponential. Choose u as the first type in ILATE that appears in the integrand.
🔄
Reversing Signs for ∫sin x dx
∫sin x dx = −cos x + C, NOT +cos x. The minus sign is frequently dropped.
❌ Wrong
∫sin x dx = cos x + C ✗
(missing minus sign)
✓ Correct
∫sin x dx = −cos x + C ✓
(verify: d/dx[−cosx]
= −(−sinx) = sinx ✓)
Memory check: differentiation and integration are inverses. d/dx[cos x] = −sin x, so ∫(−sin x) dx = cos x, therefore ∫sin x dx = −cos x. Always verify by differentiating the answer.
Chapter Intelligence
Integration is the crown jewel of calculus — master it and 10+ marks become accessible.
EAPCET Weightage (2019–2024)
Standard integrals (direct formula)~9 Definite integrals & properties~7 High-Yield PYQ Patterns
∫f'(x)/f(x) dx = ln|f(x)|Substitution: ∫f(g(x))g'(x)dx∫x eˣ dx by partsKing's property for trig integralsArea between parabola and lineEvaluate definite ∫ using limits∫eˣ[f(x)+f'(x)]dx pattern
Exam Strategy
- Pattern recognition is key. When numerator is derivative of denominator → ln form. When integrand = f(g)×g' → substitution. When product of different function types → IBP.
- King's property (f(a-x) substitution) solves many definite integral problems in one step — especially ∫sin/(sin+cos) type integrals.
- eˣ[f(x)+f'(x)] pattern: immediately write eˣf(x)+C. This appears frequently and saves 3+ steps.
- Area questions: set up the definite integral carefully, check if curve crosses x-axis in the interval (split if it does), apply absolute values.
- Differentiation + Integration together = the single most point-rich section of EAPCET Maths. Master these two chapters for maximum score impact.