Integral Calculus - Formulas & Rules
IB Mathematics Analysis & Approaches (SL & HL)
📐 Indefinite Integrals (Antiderivatives)
Definition:
An indefinite integral represents the family of all antiderivatives of a function. The result includes a constant of integration \(C\).
Notation:
\[\int f(x)\,dx = F(x) + C\]
where \(F'(x) = f(x)\) and \(C\) is the constant of integration
⚡ Power Rule for Integration
General Power Rule:
\[\int x^n\,dx = \frac{x^{n+1}}{n+1} + C\]
where \(n \neq -1\), \(n \in \mathbb{R}\)
Given in formula booklet
With Coefficient:
\[\int ax^n\,dx = a \cdot \frac{x^{n+1}}{n+1} + C\]
Special Cases:
• Constant: \(\int k\,dx = kx + C\)
• Linear: \(\int x\,dx = \frac{x^2}{2} + C\)
• Square: \(\int x^2\,dx = \frac{x^3}{3} + C\)
• Negative power: \(\int x^{-2}\,dx = -x^{-1} + C = -\frac{1}{x} + C\)
➕ Basic Integration Rules
Constant Multiple Rule:
\[\int kf(x)\,dx = k\int f(x)\,dx\]
where \(k\) is a constant
Sum/Difference Rule:
\[\int [f(x) \pm g(x)]\,dx = \int f(x)\,dx \pm \int g(x)\,dx\]
Integrate term by term
📊 Standard Integrals
Trigonometric Functions:
\[\int \sin x\,dx = -\cos x + C\]
\[\int \cos x\,dx = \sin x + C\]
\[\int \sec^2 x\,dx = \tan x + C\]
Given in formula booklet (angles in radians)
Exponential Function:
\[\int e^x\,dx = e^x + C\]
Given in formula booklet
Reciprocal Function:
\[\int \frac{1}{x}\,dx = \ln|x| + C\]
where \(x \neq 0\)
Given in formula booklet
🔗 Integration of Linear Composite Functions
General Rule:
For \(f(ax + b)\), divide by the coefficient of \(x\)
Power Function:
\[\int (ax + b)^n\,dx = \frac{(ax + b)^{n+1}}{a(n+1)} + C\]
where \(n \neq -1\), \(a \neq 0\)
Trigonometric:
\[\int \sin(ax + b)\,dx = -\frac{1}{a}\cos(ax + b) + C\]
\[\int \cos(ax + b)\,dx = \frac{1}{a}\sin(ax + b) + C\]
Exponential:
\[\int e^{ax + b}\,dx = \frac{1}{a}e^{ax + b} + C\]
Reciprocal:
\[\int \frac{1}{ax + b}\,dx = \frac{1}{a}\ln|ax + b| + C\]
🎯 Definite Integrals
Fundamental Theorem of Calculus:
\[\int_a^b f(x)\,dx = F(b) - F(a)\]
where \(F'(x) = f(x)\), \(a\) = lower limit, \(b\) = upper limit
Given in formula booklet
Alternative Notation:
\[\int_a^b f(x)\,dx = \left[F(x)\right]_a^b\]
Important Properties:
• No constant \(C\) needed for definite integrals
• \(\int_a^a f(x)\,dx = 0\) (same limits)
• \(\int_a^b f(x)\,dx = -\int_b^a f(x)\,dx\) (swap limits)
• \(\int_a^b f(x)\,dx + \int_b^c f(x)\,dx = \int_a^c f(x)\,dx\) (split intervals)
📏 Area Under a Curve
Area Above x-axis:
\[\text{Area} = \int_a^b f(x)\,dx\]
When \(f(x) \geq 0\) on \([a, b]\)
Area Below x-axis:
\[\text{Area} = -\int_a^b f(x)\,dx = \int_a^b |f(x)|\,dx\]
When \(f(x) \leq 0\) on \([a, b]\), take absolute value
Mixed (Above & Below):
Split at x-intercepts and calculate each region separately, taking absolute values
📐 Area Between Two Curves
General Formula:
\[\text{Area} = \int_a^b [f(x) - g(x)]\,dx\]
where \(f(x) \geq g(x)\) on \([a, b]\)
(upper function minus lower function)
Method:
1. Find intersection points to determine limits
2. Identify which function is above the other
3. Integrate (upper - lower)
4. If curves cross, split into regions
🔄 Integration by Substitution (HL)
Method (Indefinite Integral):
1. Choose substitution \(u = g(x)\)
2. Find \(\frac{du}{dx}\), then \(dx = \frac{du}{du/dx}\)
3. Substitute into integral
4. Integrate with respect to \(u\)
5. Substitute back to express in terms of \(x\)
Method (Definite Integral):
• Change limits: when \(x = a\), find \(u = g(a)\); when \(x = b\), find \(u = g(b)\)
• No need to substitute back
• Integrate directly with new limits
✖️ Integration by Parts (HL)
Formula:
\[\int u\,dv = uv - \int v\,du\]
Given in formula booklet
Alternative Form:
\[\int u(x)v'(x)\,dx = u(x)v(x) - \int u'(x)v(x)\,dx\]
Choosing u and dv (LIATE Rule):
Priority for \(u\) (first that appears):
Logarithmic → Inverse trig → Algebraic → Trigonometric → Exponential
🌀 Volume of Revolution (HL)
Rotation About x-axis:
\[V = \pi\int_a^b [f(x)]^2\,dx\]
Given in formula booklet
Rotation About y-axis:
\[V = \pi\int_c^d [g(y)]^2\,dy\]
Express \(x\) in terms of \(y\)
Concept:
Rotate a 2D area 360° around an axis to create a 3D solid. The volume is calculated by summing infinite circular disk cross-sections.
🚗 Kinematics Applications
Velocity from Acceleration:
\[v(t) = \int a(t)\,dt\]
Displacement from Velocity:
\[s(t) = \int v(t)\,dt\]
Total Distance Traveled:
\[\text{Distance} = \int_{t_1}^{t_2} |v(t)|\,dt\]
Use absolute value to account for direction changes
Relationships:
• Acceleration \(\xrightarrow{\text{integrate}}\) Velocity \(\xrightarrow{\text{integrate}}\) Displacement
• Displacement \(\xrightarrow{\text{differentiate}}\) Velocity \(\xrightarrow{\text{differentiate}}\) Acceleration
💡 Exam Tip: Most integration formulas are given in the IB formula booklet including power rule, standard integrals (sin, cos, e^x, 1/x), definite integral formula, integration by parts, and volume of revolution. Always add +C for indefinite integrals. For definite integrals, use your GDC to check answers. Remember: for linear composites f(ax+b), divide by coefficient of x. For area below x-axis, take absolute value. Integration is the reverse of differentiation - check your answers by differentiating!