Ever wondered how to “undo” a derivative and get back the original function? That's what integration does.
In simple words, integration is the reverse of differentiation. It tells you what original shape (function) could have produced a given rate of change. Think of it like adding up tiny slices of a cake to find the whole cake.
Why integration matters in CBSE Class 12
Integration shows up everywhere in the board exams – from finding areas and volumes to solving differential equations. Mastering the basic rules saves you time and stops you from getting stuck on a long calculation.
Power Rule – the easiest way to integrate xⁿ
The power rule lets you integrate any power of x (like x² or x⁻³). The rule says:
∫xⁿdx = (xⁿ⁺¹)/(n+1) + C
Here ∫ is the integral sign, dx tells you the variable, n is any real number except -1, and C is the constant of integration (the “+ something” you always add because many functions share the same derivative).
Example: ∫x³dx = (x⁴)/4 + C.
Imagine water flowing through a pipe. If the flow rate changes as x³, the power rule tells you the total amount of water that has passed after a certain distance.
Constant Multiple Rule
If a constant sits in front of a function, you can pull it out of the integral.
∫k·f(x)dx = k·∫f(x)dx
k is just a number like 5 or -2.
Example: ∫5x²dx = 5·∫x²dx = 5·(x³/3) + C = (5x³)/3 + C.
Sum and Difference Rule
Integrating a sum (or difference) of functions is the same as adding (or subtracting) their separate integrals.
∫[f(x) ± g(x)]dx = ∫f(x)dx ± ∫g(x)dx
Example: ∫(x³ + 2x)dx = ∫x³dx + ∫2xdx = (x⁴)/4 + (2·x²)/2 + C = (x⁴)/4 + x² + C.
Exponential Rule
Exponential functions look like aⁿ where a is a constant (like e, the natural base). The rule is:
∫aˣdx = (aˣ)/ln(a) + C
ln(a) is the natural logarithm of a. For the special case a = e, ln(e)=1, so the formula simplifies to ∫eˣdx = eˣ + C.
Example: ∫3ˣdx = (3ˣ)/ln3 + C.
Trigonometric Rules
These let you integrate sine, cosine and their relatives.
- ∫sin x dx = –cos x + C
- ∫cos x dx = sin x + C
- ∫sec² x dx = tan x + C
- ∫csc² x dx = –cot x + C
Example: ∫2sin x dx = –2cos x + C.
Quick Reference Table
| Rule | Formula | Example |
|---|---|---|
| Power | ∫xⁿdx = xⁿ⁺¹/(n+1)+C | ∫x⁴dx = x⁵/5 + C |
| Constant multiple | ∫k·f(x)dx = k·∫f(x)dx | ∫7cos x dx = 7sin x + C |
| Sum/Difference | ∫[f±g]dx = ∫fdx ± ∫gdx | ∫(x²–3x)dx = x³/3 – (3x²)/2 + C |
| Exponential | ∫aˣdx = aˣ/ln a + C | ∫2ˣdx = 2ˣ/ln 2 + C |
| Trig (sin) | ∫sin x dx = –cos x + C | ∫4sin x dx = –4cos x + C |
📝 Likely Exam Questions
- Find the integral of 3x⁵ – 2x + 7.
Solution: ∫(3x⁵ – 2x + 7)dx = 3·(x⁶/6) – 2·(x²/2) + 7x + C = (x⁶)/2 – x² + 7x + C. - Integrate 5eˣ with respect to x.
Solution: ∫5eˣdx = 5·eˣ + C. - Evaluate ∫(sin x + cos x)dx.
Solution: ∫sin x dx + ∫cos x dx = –cos x + sin x + C. - Find ∫4·2ˣdx.
Solution: 4·∫2ˣdx = 4·(2ˣ/ln 2) + C = (4·2ˣ)/ln 2 + C. - Calculate ∫(3sec² x – 2csc² x)dx.
Solution: 3·∫sec² x dx – 2·∫csc² x dx = 3·tan x + 2·cot x + C.