Why integration matters in everyday maths
Ever wondered how we find the area under a curve? That's integration in action. It pops up in physics, economics, even in video game graphics.
💡 In Simple Words: Integration is the reverse of differentiation. If differentiation tells you the speed from a distance, integration tells you the distance from a speed. So, you’re basically adding up tiny pieces to get the whole.
Basic integration rules you must know
These rules are the shortcuts that let you jump straight to the antiderivative (the function whose derivative gives you back the original). Think of them as the “cheat codes” for integrals.
Power rule
The power rule says ∫ x^n dx = x^(n+1)/(n+1) + C, where n ≠ -1. Here, ∫ is the integral sign, x^n is the integrand (the function you’re integrating), and C is the constant of integration (a “+ anything” that disappears when you differentiate).
Example: ∫ 3x^2 dx = 3 * (x^3/3) + C = x^3 + C.
Constant multiple rule
If a constant sits in front of the integrand, you can pull it out: ∫ k·f(x) dx = k·∫ f(x) dx.
Example: ∫ 5 sin x dx = 5·(-cos x) + C = -5 cos x + C.
Sum and difference rule
Integration works term‑by‑term: ∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx.
Example: ∫ (2x^3 + 4x) dx = ∫ 2x^3 dx + ∫ 4x dx = (2·x^4/4) + (4·x^2/2) + C = (x^4/2) + 2x^2 + C.
Exponential rule
For the natural exponential e (≈2.718), ∫ e^{ax} dx = (1/a)·e^{ax} + C, where a is a constant.
Example: ∫ e^{3x} dx = (1/3) e^{3x} + C.
Basic trigonometric rules
- ∫ sin x dx = -cos x + C
- ∫ cos x dx = sin x + C
- ∫ sec^2 x dx = tan x + C
- ∫ csc^2 x dx = -cot x + C (cot is cosine over sine)
Example: ∫ (2 sin x + 3 cos x) dx = -2 cos x + 3 sin x + C.
Quick reference table
| Rule | Formula | Example |
|---|---|---|
| Power | ∫ x^n dx = x^{n+1}/(n+1) + C | ∫ x^4 dx = x^5/5 + C |
| Constant multiple | ∫ k·f(x) dx = k·∫ f(x) dx | ∫ 7 cos x dx = 7 sin x + C |
| Sum/Difference | ∫ (f ± g) dx = ∫ f dx ± ∫ g dx | ∫ (x^2 – 5) dx = x^3/3 – 5x + C |
| Exponential | ∫ e^{ax} dx = (1/a) e^{ax} + C | ∫ e^{‑2x} dx = –½ e^{‑2x} + C |
| Trig (sin, cos) | ∫ sin x dx = –cos x + C; ∫ cos x dx = sin x + C | ∫ 4 sin x dx = –4 cos x + C |
Tips for solving CBSE/ISC integration problems
- Always add “+ C” at the end – the exam marks it.
- Check if the integrand matches a rule directly; if not, try to rewrite it.
- Watch out for the special case n = –1 in the power rule; that becomes a logarithm: ∫ 1/x dx = ln|x| + C.
- For composite functions, see if a simple substitution will turn it into one of the basic forms.
📝 Likely Exam Questions
- Find ∫ (3x^3 – 2x + 5) dx.
Answer: (3·x^4/4) – (2·x^2/2) + 5x + C = (3/4)x^4 – x^2 + 5x + C. - Evaluate ∫ 4e^{2x} dx.
Answer: 4·(1/2) e^{2x} + C = 2 e^{2x} + C. - Compute ∫ (sin x + 2 cos x) dx.
Answer: –cos x + 2 sin x + C. - State the antiderivative of 1/x and mention any condition.
Answer: ∫ 1/x dx = ln|x| + C; valid for x ≠ 0. - Using the sum rule, integrate ∫ (x^2 + e^x) dx.
Answer: x^3/3 + e^x + C.