Concept Core
From basic integrals to integration by parts and definite integral properties.
Standard Integrals β Must Know
β«xβΏ dx = xβΏβΊΒΉ/(n+1) + C (n β β1) β«1/x dx = ln|x| + C
β«eΛ£ dx = eΛ£ + C β«aΛ£ dx = aΛ£/ln a + C
β«sin x dx = βcos x + C β«cos x dx = sin x + C
β«secΒ²x dx = tan x + C β«1/β(1βxΒ²) dx = sinβ»ΒΉx + C
β«1/(1+xΒ²) dx = tanβ»ΒΉx + C β«1/(aΒ²+xΒ²) dx = (1/a)tanβ»ΒΉ(x/a)+C
Substitution Method
When integrand has f(g(x))Γg'(x): substitute u = g(x), du = g'(x)dx
Example: β«2x e^(xΒ²) dx. Let u = xΒ², du = 2x dx. β β«eα΅ du = eα΅ + C = e^(xΒ²) + C
Standard substitution results:
β«f'(x)/f(x) dx = ln|f(x)| + C
β«[f(x)]βΏ f'(x) dx = [f(x)]βΏβΊΒΉ/(n+1) + C
Integration by Parts (IBP)
β«u dv = uv β β«v du
Choose u using ILATE priority: Inverse trig β Logarithm β Algebraic β Trig β Exponential. The first type in ILATE is u, the rest is dv.
Example: β«x eΛ£ dx β u=x, dv=eΛ£ dx β uvββ«v du = xeΛ£ β β«eΛ£ dx = xeΛ£ β eΛ£ + C
Definite Integral Properties
β«βα΅ f(x)dx = ββ«α΅¦β f(x)dx
β«βα΅ f(x)dx = β«βαΆ f(x)dx + β«αΆα΅ f(x)dx
β«βα΅ f(x)dx = 0
β«βα΅ f(x)dx = β«βα΅ f(aβx)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 = β«βα΅ |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 = β«βα΅ [f(x) β g(x)] dx (f above g).
Special Integrals by Reduction
β«β(aΒ²βxΒ²) dx = (x/2)β(aΒ²βxΒ²) + (aΒ²/2)sinβ»ΒΉ(x/a) + C
β«1/β(xΒ²+aΒ²) dx = ln|x + β(xΒ²+aΒ²)| + C
β«eΛ£[f(x)+f'(x)] dx = eΛ£ f(x) + C
The last one is particularly useful β when an eΛ£ factor multiplies f(x)+f'(x), the integral is simply eΛ£ f(x).
Formula Vault
All integration formulas for EAPCET.
Power Integral
β«xβΏ dx = xβΏβΊΒΉ/(n+1) + C
n β β1; don't forget +C
Exponential
β«eΛ£ dx = eΛ£ + C
β«aΛ£ dx = aΛ£/ln(a) + C
Reciprocal
β«1/x dx = ln|x| + C
Absolute value important
sin and cos
β«sin x dx = βcos x + C
β«cos x dx = sin x + C
secΒ² and cosecΒ²
β«secΒ²x dx = tan x + C
β«cosecΒ²x dx = βcot x + C
Inverse Trig
β«1/β(1βxΒ²) dx = sinβ»ΒΉx + C
β«1/(1+xΒ²) dx = tanβ»ΒΉx + C
Logarithmic Form
β«f'(x)/f(x) dx = ln|f(x)| + C
Numerator = derivative of denominator
Integration by Parts
β«u dv = uv β β«v du
ILATE rule for choosing u
King's Property
β«βα΅ f(x)dx = β«βα΅ f(aβx)dx
Extremely useful for definite integrals
eΛ£ Composite
β«eΛ£[f(x)+f'(x)]dx = eΛ£f(x)+C
Recognise this pattern for instant answer
Worked Examples
5 problems β substitution, by parts, definite integral, King's property, and a trap.
EasyIntegrate β«(3xΒ² + 2x β 5)dxβΎ
Find β«(3xΒ² + 2x β 5) dx.
1
Apply power rule to each term:
2
= 3xΒ³/3 + 2xΒ²/2 β 5x + C = xΒ³ + xΒ² β 5x + C
β β« = xΒ³ + xΒ² β 5x + C
EasyFind β«(2x/(xΒ²+1))dxβΎ
Evaluate β«2x/(xΒ²+1) dx.
1
Numerator 2x is the derivative of denominator (xΒ²+1).
2
Using β«f'/f dx = ln|f| + C:
3
= ln|xΒ²+1| + C = ln(xΒ²+1) + C (xΒ²+1 always positive)
β β« = ln(xΒ²+1) + C
MediumIntegrate β«x eΛ£ dx by partsβΎ
Evaluate β«x eΛ£ dx using integration by parts.
1
ILATE: u = x (Algebraic), dv = eΛ£ dx (Exponential)
3
β«x eΛ£ dx = uv β β«v du = xeΛ£ β β«eΛ£ dx = xeΛ£ β eΛ£ + C = eΛ£(xβ1) + C
β β«x eΛ£ dx = eΛ£(xβ1) + C
EAPCET LevelUse King's property: β«β^(Ο/2) sin x/(sin x + cos x) dxβΎ
Evaluate β«β^(Ο/2) sin x/(sin x + cos x) dx.
1
Let I = β«β^(Ο/2) sinx/(sinx+cosx) dx
2
By King's property: f(x) β f(Ο/2βx): sin(Ο/2βx)=cosx, cos(Ο/2βx)=sinx
3
I = β«β^(Ο/2) cosx/(cosx+sinx) dx
4
Add both integrals: 2I = β«β^(Ο/2) (sinx+cosx)/(sinx+cosx) dx = β«β^(Ο/2) 1 dx = Ο/2
β I = Ο/4 (King's property approach)
Trap Questionβ«β^1 x/(1+xΒ²) dx β students add constant C to definite integralβΎ
Evaluate the definite integral β«βΒΉ x/(1+xΒ²) dx. β οΈ Classic formatting trap.
1
The trap: Adding arbitrary constant C to a definite integral. Definite integrals have a specific numerical value β no C.
2
Let u = 1+xΒ², du = 2x dx β x dx = du/2
3
= (1/2)β«βΒ² 1/u du = (1/2)[ln u]βΒ² = (1/2)(ln 2 β ln 1) = (1/2) ln 2
β β«βΒΉ x/(1+xΒ²) dx = (ln 2)/2 β no +C for definite integrals
Mistake DNA
4 integration errors from EAPCET distractor analysis.
β
Adding +C to Definite Integrals
Definite integrals give a specific numerical value. The constant of integration appears only in indefinite integrals.
β Wrong
β«βΒΉ x dx = [xΒ²/2 + C]βΒΉ
= (1/2+C) β (0+C) β
(C cancels anyway but
adding it is wrong form)
β Correct
β«βΒΉ x dx = [xΒ²/2]βΒΉ
= 1/2 β 0 = 1/2 β
No +C for definite β
Indefinite integral: f(x) + C (unknown constant; family of curves). Definite integral: specific number between limits. C is not needed for definite integrals.
π’
Forgetting n+1 in Denominator for Power Rule
β«xβΏ dx = xβΏβΊΒΉ/(n+1) + C. Students raise the power but forget to divide by the new exponent.
β Wrong
β«xΒ³ dx = xβ΄ + C β
(forgot to divide by 4)
β Correct
β«xΒ³ dx = xβ΄/4 + C β
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.
π
ILATE: Choosing Wrong Function as u in IBP
Choosing a poor u leads to a more complex integral. ILATE gives the optimal choice.
β Wrong
β«x sin x dx: u=sin x,
dv=x dx β
β makes integral harder
β Correct
ILATE: A before T β
u=x (Algebraic)
dv=sinx dx β
β simpler integral
ILATE order: Inverse trig, Logarithm, Algebraic, Trigonometric, Exponential. Choose u as the first type in ILATE that appears in the integrand.
π
Reversing Signs for β«sin x dx
β«sin x dx = βcos x + C, NOT +cos x. The minus sign is frequently dropped.
β Wrong
β«sin x dx = cos x + C β
(missing minus sign)
β Correct
β«sin x dx = βcos x + C β
(verify: d/dx[βcosx]
= β(βsinx) = sinx β)
Memory check: differentiation and integration are inverses. d/dx[cos x] = βsin x, so β«(βsin x) dx = cos x, therefore β«sin x dx = βcos x. Always verify by differentiating the answer.
Chapter Intelligence
Integration is the crown jewel of calculus β master it and 10+ marks become accessible.
EAPCET Weightage (2019β2024)
Standard integrals (direct formula)~9 Definite integrals & properties~7 High-Yield PYQ Patterns
β«f'(x)/f(x) dx = ln|f(x)|Substitution: β«f(g(x))g'(x)dxβ«x eΛ£ dx by partsKing's property for trig integralsArea between parabola and lineEvaluate definite β« using limitsβ«eΛ£[f(x)+f'(x)]dx pattern
Exam Strategy
- Pattern recognition is key. When numerator is derivative of denominator β ln form. When integrand = f(g)Γg' β substitution. When product of different function types β IBP.
- King's property (f(a-x) substitution) solves many definite integral problems in one step β especially β«sin/(sin+cos) type integrals.
- eΛ£[f(x)+f'(x)] pattern: immediately write eΛ£f(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.
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