Integrals
1. Indefinite Integral - Anti-Derivative
Integration is reverse of differentiation. If derivative of F(x) is f(x) then integral of f(x) dx = F(x)+C. Example derivative of x² is 2x so integral of 2x dx = x²+C. C is constant of integration because derivative of constant 0, so infinite family of curves shifted up/down all have same derivative. Indefinite integral gives general family, no limits. Symbol ∫ is elongated S for sum.
If F'(x)=f(x) then ∫f(x)dx=F(x)+C. Anti-derivative. Example F=x² F'=2x so ∫2x dx=x²+C. +C because derivative of constant 0, infinite antiderivatives shifted up/down. Indefinite no limits, family of functions.
• Reverse of derivative: If F'=f then ∫f dx=F+C
• Example: d/dx[x²]=2x so ∫2x dx=x²+C
• +C constant because derivative of constant 0
• Family of curves shifted up/down
• Indefinite = general, no limits
• Symbol ∫ elongated S for sum
2. Substitution Method - u-sub
When integral has form ∫f(g(x))·g'(x) dx, put u=g(x) then du=g'(x)dx, integral becomes ∫f(u)du easy. Steps: choose u inner function, find du, rewrite integral in u, integrate, substitute back to x. Example ∫2x cos(x²)dx let u=x² du=2x dx → ∫cos u du=sin u+C=sin(x²)+C. Example ∫3x²/(x³+1)dx u=x³+1 du=3x²dx → ∫1/u du=ln|u|+C. Always replace back to x. Look for function and its derivative together.
u-sub: Put u=g(x) du=g'(x)dx ∫f(g(x))g'(x)dx=∫f(u)du. Steps: choose u inner, find du, rewrite in u, integrate, sub back. Example 2x cos(x²) u=x² → sin(x²)+C. Look for function and its derivative multiplied.
• Idea: ∫f(g(x))·g'(x)dx let u=g(x) du=g'(x)dx → ∫f(u)du
• Steps: Choose u → Find du → Rewrite in u → Integrate → Sub back to x
• Example 2x cos(x²): u=x² du=2x dx → sin(x²)+C
• Example 3x²/(x³+1): u=x³+1 → ln|x³+1|+C
• Tip: Make sure du present adjust constant factor
• Look for inner function and its derivative
3. Partial Fractions - Split Big Fraction
When integral is rational function P(x)/Q(x) where degree P
Partial fractions: Rational P/Q deg P
• Use when rational P/Q deg P
• Repeated (x-a)² → A/(x-a)+B/(x-a)²
• Quadratic (x²+...) → (Ax+B)/quadratic
• Example (2x+3)/((x+1)(x+2))=A/(x+1)+B/(x+2)
• Solve A,B then integrate each to ln
• Then ∫ becomes sum of easy logs
4. Integration by Parts - ILATE Rule
Integration by parts: ∫u dv = uv - ∫v du, used when product of two functions where one easy to differentiate other easy to integrate. ILATE rule to choose u: Inverse trig, Logarithmic, Algebraic, Trigonometric, Exponential priority for u. Example ∫x e^x dx: u=x (algebraic) dv=e^x dx → du=dx v=e^x → uv - ∫v du = x e^x - ∫e^x dx = x e^x - e^x +C = e^x(x-1)+C. Example ∫ln x dx: u=ln x dv=dx. If ∫ln x alone, take u=ln x dv=dx.
By parts ∫u dv=uv-∫v du. ILATE to choose u: Inverse, Log, Algebraic, Trig, Exponential. u first in ILATE, dv remaining. Example ∫x e^x u=x dv=e^x → x e^x - e^x+C. ∫ln x dx u=ln x dv=dx.
• Formula: ∫u dv = uv - ∫v du
• ILATE priority for u: I-Inverse, L-Log, A-Algebraic, T-Trig, E-Exponential
• u chosen first in ILATE, remaining dv
• Example x e^x: u=x dv=e^x → x e^x - e^x+C=e^x(x-1)+C
• Example ln x: u=ln x dv=dx → x ln x - x +C
• If single log or inverse, dv=dx
5. Special Integrals - sqrt Forms
Some standard forms directly: ∫dx/√(x²+a²)=ln|x+√(x²+a²)|+C, ∫dx/√(x²-a²)=ln|x+√(x²-a²)|+C, ∫dx/√(a²-x²)=sin⁻¹(x/a)+C, ∫√(x²+a²)dx= x/2 √(x²+a²) + a²/2 ln|x+√(x²+a²)|+C, ∫√(a²-x²)dx= x/2 √(a²-x²)+a²/2 sin⁻¹(x/a)+C. These come from trig substitution x=a tanθ or sinθ. Remember them, direct formula saves time.
Special: ∫dx/√(x²±a²)=ln|x+√(x²±a²)|, ∫dx/√(a²-x²)=sin⁻¹(x/a), ∫√(x²+a²)dx=x/2√(...)+a²/2 ln|...|, ∫√(a²-x²)dx=x/2√(...)+a²/2 sin⁻¹(x/a). From trig sub x=a tan or sin.
• ∫dx/√(x²+a²)=ln|x+√(x²+a²)|+C
• ∫dx/√(x²-a²)=ln|x+√(x²-a²)|+C
• ∫dx/√(a²-x²)=sin⁻¹(x/a)+C
• ∫√(x²+a²)dx= x/2√(x²+a²)+a²/2 ln|x+√(...)|+C
• ∫√(a²-x²)dx= x/2√(a²-x²)+a²/2 sin⁻¹(x/a)+C
• From trig sub x=a tanθ or a sinθ
• Direct formula remember
6. Definite Integral - Area Under Curve
Definite integral ∫_a^b f(x)dx is area under curve from x=a to b, with limits. If f(x)≥0 area positive, if below x-axis negative. It is limit of sum of rectangles. Geometrically area. Example ∫_0^1 x² dx = [x³/3]_0^1=1/3 area under parabola from 0 to1. Definite has value, no +C. If you know indefinite F(x), definite = F(b)-F(a). Definite integral as limit of sum: ∫_a^b f(x)dx = lim n→∞ sum f(xi)Δx.
Definite ∫_a^b f(x)dx = area under curve a to b, limit of sum rectangles. Value not family. If F'=f then ∫_a^b f=F(b)-F(a). Example ∫0^1 x²=[x³/3]0^1=1/3 area. No +C, definite number.
• ∫_a^b f(x)dx = area under curve from a to b
• Limit of sum of rectangles Σ f(xi)Δx
• If f≥0 area positive, if below negative
• Value not family, no +C
• If F'=f then definite = F(b)-F(a)
• Example ∫0^1 x² = [x³/3]0^1=1/3
• Geometrically area, numerically number
7. Fundamental Theorem and Properties
FTC1: If F(x)=∫_a^x f(t)dt then F'(x)=f(x). FTC2: If F'=f then ∫_a^b f(x)dx=F(b)-F(a) evaluates definite. Properties of definite: ∫_a^b f = -∫_b^a f reverse sign, ∫_a^a=0, ∫_a^b f = ∫_a^c f + ∫_c^b f split, ∫_0^a f(x)dx=∫_0^a f(a-x)dx King property, even-odd: If f even ∫_-a^a f=2∫_0^a f, if odd ∫_-a^a=0. King property very useful for solving tricky integrals.
FTC1 F(x)=∫_a^x f(t)dt ⇒ F'=f. FTC2 ∫_a^b f=F(b)-F(a). Properties: ∫_a^b=-∫_b^a, ∫_a^a=0, split ∫_a^c+∫_c^b, King ∫_0^a f(x)=∫_0^a f(a-x), even f(-x)=f(x) ∫_-a^a=2∫_0^a, odd f(-x)=-f(x) ∫_-a^a=0.
• FTC1: F(x)=∫_a^x f(t)dt → F'(x)=f(x)
• FTC2: ∫_a^b f(x)dx=F(b)-F(a)
• ∫_a^b = -∫_b^a, ∫_a^a=0
• Split: ∫_a^b=∫_a^c+∫_c^b
• King: ∫_0^a f(x)dx=∫_0^a f(a-x)dx very useful
• Even: f(-x)=f(x) ∫_-a^a=2∫_0^a f
• Odd: f(-x)=-f(x) ∫_-a^a=0
8. Real Life Applications - Where Integrals Live
Integrals give area under curve, volume of solids, work done, centre of mass. Engineering: area of irregular shapes, volume of dam water. Physics: distance from velocity integral, work = ∫F dx. Economics: consumer surplus area. Medical: blood flow. Definite integral area between curves. In real life GPS area calculation uses integrals. Rocket fuel consumption integral of rate. Population growth integral.
Real: Area irregular shapes, volume dam water, work ∫F dx, distance from velocity, consumer surplus, blood flow, area between curves, GPS area, rocket fuel, population growth, centre of mass.
• Area: Irregular land area, dam volume
• Physics: Distance = ∫velocity dt, Work=∫F dx
• Engineering: Volume solids, centre of mass
• Economics: Consumer surplus area
• Medical: Blood flow rate integral
• Between curves: Area = ∫(top-bottom)dx
• Daily: GPS polygon area, rocket fuel consumption, population growth
• Indefinite: ∫f dx=F+C if F'=f, family +C
• Substitution: u=g(x) du=g' dx ∫f(g)g' dx=∫f(u)du, look for function and derivative
• Partial Fractions: Rational P/Q split linear→A/(x-a), repeated→A/(x-a)+B/(x-a)², quadratic→(Ax+B)/quad → logs
• By Parts: ∫u dv=uv-∫v du, ILATE I-L-A-T-E for u, e.g. x e^x → e^x(x-1)+C
• Special sqrt: dx/√(x²+a²)=ln|x+√| , dx/√(a²-x²)=sin⁻¹(x/a), √ forms half x root + half a² ln/sin⁻¹
• Definite: ∫_a^b f = area a to b, =F(b)-F(a), no +C, limit of sum
• FTC: F(x)=∫_a^x f(t)dt → F'=f, ∫_a^b f=F(b)-F(a)
• Properties: ∫_a^b=-∫_b^a, ∫_a^a=0, split, King ∫_0^a f(x)=∫_0^a f(a-x), even 2∫_0^a odd 0
• Real: Area irregular, volume, work ∫F dx, distance ∫v dt, consumer surplus, GPS area