Ever wondered how you can find the area under a curve without drawing a million tiny rectangles? That’s the magic of integration – the flip side of differentiation.
💡 In Simple Words: Integration is like adding up an infinite number of tiny pieces to get a whole. If you think of a curve as a wiggly road, integration tells you the total distance you’d travel if you could walk every little step at once.
Key Integration Rules Every CBSE Student Should Master
1. Power Rule
Rule: ∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C, where n ≠ –1.
Why it works: Imagine stacking blocks of height xⁿ; the total height grows a little faster, so we add 1 to the exponent and divide by that new number.
Example: ∫x³ dx = (x⁴)/4 + C.
2. Constant Multiple Rule
Rule: ∫k·f(x) dx = k·∫f(x) dx, where k is a constant (a fixed number).
Think of k as a scaling factor – if you double every slice of a pizza, the whole pizza’s area doubles.
Example: ∫5·sin x dx = 5·(–cos x) + C = –5cos x + C.
3. Sum/Difference Rule
Rule: ∫[f(x) ± g(x)] dx = ∫f(x) dx ± ∫g(x) dx.
It’s like adding two separate streams of water; the total flow is just the sum of each stream.
Example: ∫(x² + 3x) dx = ∫x² dx + ∫3x dx = (x³)/3 + (3x²)/2 + C.
4. Basic Trigonometric Integrals
- ∫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 (rate of change) of cosine is –sine, and so on.
5. Exponential and Logarithmic Integrals
- ∫eˣ dx = eˣ + C (e is the natural base ≈2.718)
- ∫aˣ dx = (aˣ)/(ln a) + C, where a>0, a≠1 (ln is the natural logarithm)
- ∫(1/x) dx = ln|x| + C (ln|x| means natural log of the absolute value of x)
Quick Reference Table
| Rule | Formula | Example |
|---|---|---|
| Power Rule | ∫xⁿ dx = xⁿ⁺¹/(n+1) + C | ∫x² dx = x³/3 + C |
| Constant Multiple | ∫k·f(x) dx = k·∫f(x) dx | ∫4cos x dx = 4sin x + C |
| Sum/Difference | ∫[f±g] dx = ∫f dx ± ∫g dx | ∫(sin x – cos x) dx = –cos x – sin x + C |
| Trig Basics | ∫sin x dx = –cos x + C | ∫sin x dx = –cos x + C |
| Exponential | ∫eˣ dx = eˣ + C | ∫eˣ dx = eˣ + C |
How to Tackle Integration Problems in the Exam
1. Identify which rule matches the integrand (the function you’re integrating).
2. Check if a constant factor can be pulled out.
3. If the expression is a sum or difference, split it.
4. Apply the rule, then don’t forget the “+ C” – the constant of integration that reminds us there are many antiderivatives.
📝 Likely Exam Questions
- Find the integral of 3x⁴ – 2x + 5.
Answer: ∫(3x⁴ – 2x + 5) dx = (3·x⁵)/5 – x² + 5x + C. - Evaluate ∫cos x dx.
Answer: sin x + C. - Integrate (2eˣ – 4/x) dx.
Answer: 2eˣ – 4ln|x| + C. - Using the power rule, find ∫(1/√x) dx.
Answer: Write as x^(–½). Integral = x^(½)/(½) + C = 2√x + C. - Compute ∫(sin x + 3) dx.
Answer: –cos x + 3x + C.