Ever wondered how to find the area under a curve without drawing a million tiny rectangles?
Think of an integral as a magical way to add up an infinite number of tiny pieces—like counting every grain of sand on a beach—to get the total area, distance, or volume.
What is an Integral?
An integral is the reverse of a derivative. If a derivative tells you how fast something changes, an integral tells you the total amount that has changed. In plain words, it’s the process of adding up infinitely many tiny slices. The result of an integral is often called an antiderivative—a function whose derivative gives you back the original expression.
Basic Integration Rules
Just like you have shortcut formulas for adding numbers, calculus gives you shortcut formulas for integrals. Below are the most common rules you’ll see in ISC exams.
Power Rule
For any real number n ≠ -1, the integral of xⁿ is xⁿ⁺¹/(n+1) plus a constant C. Think of it as the opposite of the power rule for differentiation.
Example: ∫ x³ dx = x⁴/4 + C.
Constant Multiple Rule
If a constant k multiplies a function, you can pull k out of the integral. It’s like factoring out a common factor before adding numbers.
Example: ∫ 5 sin x dx = 5 ∫ sin x dx = -5 cos x + C.
Sum/Difference Rule
The integral of a sum (or difference) of functions equals the sum (or difference) of their integrals. You can split a big problem into smaller, easier pieces.
Example: ∫ (2x² + 3cos x) dx = ∫ 2x² dx + ∫ 3cos x dx = 2·x³/3 + 3·sin x + C.
Exponential Rule
The integral of eˣ (where e is the natural base ≈ 2.718) is just eˣ plus C. Exponential functions behave nicely under integration.
Example: ∫ eˣ dx = eˣ + C.
Basic Trigonometric Rules
- ∫ sin x dx = -cos x + C
- ∫ cos x dx = sin x + C
- ∫ sec² x dx = tan x + C
- ∫ csc² x dx = -cot x + C
These come from the fact that the derivative of cosine is negative sine, and so on.
Quick Reference Table
| Rule | Formula | When to Use |
|---|---|---|
| Power | ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n≠-1) | Any polynomial term |
| Constant Multiple | ∫ k·f(x) dx = k·∫ f(x) dx | Factor out numbers |
| Sum/Difference | ∫ [f(x)±g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx | Split complex expressions |
| Exponential | ∫ eˣ dx = eˣ + C | Pure eˣ terms |
| Sin/Cos | ∫ sin x dx = -cos x + C; ∫ cos x dx = sin x + C | Trigonometric integrands |
How to Tackle ISC Integration Problems
- Identify the type of function (polynomial, exponential, trig).
- Check if a constant factor can be pulled out.
- Look for sums or differences you can split.
- Apply the power rule first; if the exponent is -1, remember ∫ 1/x dx = ln|x| + C (log rule).
- Verify by differentiating your answer; you should get back the original integrand.
📝 Likely Exam Questions
- Find the indefinite integral: ∫ (3x⁴ - 2x + 5) dx.
Answer: (3/5)x⁵ - x² + 5x + C. - Evaluate: ∫ 4eˣ dx.
Answer: 4eˣ + C. - Integrate the trigonometric function: ∫ 2sin x cos x dx.
Answer: Use the identity sin2x = 2sin x cos x, so ∫ sin2x dx = -½cos2x + C. - Find the antiderivative of: ∫ (1/x + x³) dx.
Answer: ln|x| + x⁴/4 + C. - Solve: ∫ (5 sec² x - 3) dx.
Answer: 5tan x - 3x + C.