Introduction to Integration
Welcome to the world of Integration! If you have already mastered Differentiation, you are halfway there. Think of Integration as the "undo" button on your computer. While Differentiation helps us find the gradient (the rate of change), Integration helps us work backwards to find the original function.
In this chapter, we will learn how to reverse the process of differentiation, find the area under curves, and solve problems involving moving objects (Kinematics). Don't worry if it feels a bit abstract at first—once you see the patterns, it becomes a very logical and satisfying puzzle!
Quick Note: This chapter belongs to the Calculus section. It is closely linked to Differentiation, so having your differentiation rules handy will be very helpful!
1. Integration as the Reverse of Differentiation
When we differentiate \(y = x^2 + 5\), we get \(\frac{dy}{dx} = 2x\).
When we differentiate \(y = x^2 - 10\), we also get \(\frac{dy}{dx} = 2x\).
Because the constant (the number without an \(x\)) disappears during differentiation, we must add a constant of integration, represented by \(C\), whenever we integrate.
The symbol for integration is \(\int\). We write it like this:
\(\int f'(x) dx = f(x) + C\)
Key Takeaway: Always remember the \(+ C\) for indefinite integrals (integrals without numbers at the top and bottom of the sign)!
2. The Basic Integration Rules
A. The Power Rule
To integrate \(x^n\), we follow two simple steps:
1. Add 1 to the power.
2. Divide by the new power.
\(\int x^n dx = \frac{x^{n+1}}{n+1} + C\) (where \(n\) is any rational number except \(-1\))
Example: \(\int x^3 dx = \frac{x^4}{4} + C\)
B. Trigonometric Functions
These come directly from reversing our differentiation rules. Remember that only \(\sin\), \(\cos\), and \(\sec^2\) are required for integration in this syllabus:
1. \(\int \sin x dx = -\cos x + C\)
2. \(\int \cos x dx = \sin x + C\)
3. \(\int \sec^2 x dx = \tan x + C\)
Memory Aid: In differentiation, \(\cos\) becomes negative \(\sin\). In integration, it is the opposite: \(\sin\) becomes negative \(\cos\)!
C. Exponential Functions
The function \(e^x\) is special because it stays the same!
\(\int e^x dx = e^x + C\)
D. Constant Multiples and Sums
Just like in differentiation, constants stay where they are, and you can integrate terms one by one if they are separated by plus or minus signs.
\(\int [f(x) + g(x)] dx = \int f(x) dx + \int g(x) dx\)
\(\int k f(x) dx = k \int f(x) dx\)
3. Integrating Linear Composites: The \((ax + b)\) Form
Sometimes, instead of just \(x\), we have a linear expression like \((2x + 3)\). When integrating these, we still follow the basic rules, but we must divide by the coefficient of \(x\) (the number \(a\)).
The Rules:
1. Power: \(\int (ax + b)^n dx = \frac{(ax + b)^{n+1}}{a(n + 1)} + C\)
2. Sine: \(\int \sin(ax + b) dx = -\frac{1}{a} \cos(ax + b) + C\)
3. Cosine: \(\int \cos(ax + b) dx = \frac{1}{a} \sin(ax + b) + C\)
4. Exponential: \(\int e^{ax + b} dx = \frac{1}{a} e^{ax + b} + C\)
Example: To integrate \(\int \cos(3x + 1) dx\), the coefficient of \(x\) is \(3\). So, the answer is \(\frac{1}{3}\sin(3x + 1) + C\).
Common Mistake: This "divide by \(a\)" trick only works if the expression inside is linear (like \(3x + 1\)). If it is \(x^2\), you cannot use this method!
4. Definite Integrals
A definite integral has upper and lower limits (numbers at the top and bottom of the \(\int\) sign). These integrals result in a specific number rather than a function with \(+ C\).
How to evaluate:
1. Integrate the function as usual (leave out the \(+ C\)).
2. Write the result in square brackets with the limits on the right: \([F(x)]_a^b\).
3. Substitute the top number, then subtract the result of substituting the bottom number.
\(\int_a^b f(x) dx = [F(x)]_a^b = F(b) - F(a)\)
Step-by-Step Example:
Evaluate \(\int_1^2 2x dx\).
1. Integrate \(2x\): we get \(x^2\).
2. Setup: \([x^2]_1^2\).
3. Calculate: \((2)^2 - (1)^2 = 4 - 1 = 3\).
5. Finding the Area Under a Curve
One of the most powerful uses of integration is finding the area of the region between a curve and the axes.
A. Area between a curve and the \(x\)-axis
The area \(A\) bounded by the curve \(y = f(x)\), the \(x\)-axis, and the vertical lines \(x = a\) and \(x = b\) is given by:
\(A = \int_a^b y dx\)
B. Regions Below the \(x\)-axis
If the region lies below the \(x\)-axis, the integral will give you a negative value. Since area cannot be negative, you must take the absolute value (modulus) of the result.
Pro-tip: If a question asks for the total area and part of the curve is above the axis while another part is below, calculate the two areas separately and add their positive values together!
Important Exclusion: You are not required to find the area between two curves for this syllabus. Focus on the area between a curve and a straight line or the axes.
6. Kinematics: Moving in a Straight Line
Integration allows us to move "up" the chain of motion:
Acceleration (\(a\)) \(\to\) Velocity (\(v\)) \(\to\) Displacement (\(s\))
1. To find Velocity, integrate Acceleration: \(v = \int a dt\)
2. To find Displacement, integrate Velocity: \(s = \int v dt\)
Did you know? In Kinematics, we usually integrate with respect to time (\(t\)). When you integrate to find \(v\) or \(s\), you often use given conditions (like "starts from rest", which means \(v=0\) at \(t=0\)) to find the value of the constant \(C\).
Summary Checklist
Quick Review:
- Did you add \(+ C\) for indefinite integrals?
- Did you divide by \(a\) for \((ax + b)\) forms?
- For area below the \(x\)-axis, did you remember to change the negative sign to positive?
- Are you using radians for all trigonometric integration? (Standard convention in Calculus!)
Common Pitfalls to Avoid:
- The "Power Rule" Trap: Don't try to integrate \(\frac{1}{x}\) using the power rule—it results in division by zero! (Note: \(\int \frac{1}{x} dx\) is not required in this syllabus).
- The Trigo Mix-up: Double-check your signs! \(\int \sin x dx = -\cos x\), but \(\frac{d}{dx} \sin x = \cos x\).
- Missing Limits: In definite integrals, always plug in the upper limit FIRST, then subtract the lower limit.