Derivatives tell you the slope. Integrals do the reverse — they accumulate. If the derivative of position is velocity, the integral of velocity is position. If the derivative gives you the rate, the integral gives you the total.
Here's how to calculate integrals from scratch.
The Power Rule for Integration
The reverse of the derivative power rule:
∫xⁿ dx = xⁿ⁺¹/(n+1) + C
Translation: increase the exponent by 1, divide by the new exponent, add +C.
Step-by-Step Example
Find ∫(3x² + 2x − 5) dx
| Term | Apply Power Rule | Result | |------|-----------------|--------| | 3x² | 3 · x³/3 | x³ | | 2x | 2 · x²/2 | x² | | −5 | −5 · x¹/1 | −5x |
∫(3x² + 2x − 5) dx = x³ + x² − 5x + C
The +C is the constant of integration — it represents all possible antiderivatives.
Why +C?
The derivative of any constant is 0. So the antiderivative of x² could be x³/3 + 5, x³/3 − 100, or x³/3 + any number. Without +C, you're missing all those possibilities.
For indefinite integrals, always add +C.
Definite Integrals — Area Under a Curve
A definite integral calculates the exact area between the curve and the x-axis, from x = a to x = b.
∫[a,b] f(x) dx = F(b) − F(a)
No +C needed — it cancels out when you subtract.
Example: ∫[0,3] (3x² + 2x − 5) dx
- Antiderivative: F(x) = x³ + x² − 5x
- F(3) = 27 + 9 − 15 = 21
- F(0) = 0 + 0 − 0 = 0
- Area = 21 − 0 = 21 square units
Common Integration Formulas
| Function | Indefinite Integral | |----------|-------------------| | xⁿ | xⁿ⁺¹/(n+1) + C | | c (constant) | cx + C | | eˣ | eˣ + C | | 1/x | ln|x| + C | | sin(x) | −cos(x) + C | | cos(x) | sin(x) + C | | sec²(x) | tan(x) + C |
The Trench Truth: The #1 integration mistake: forgetting to divide by the new exponent. ∫x³ dx = x⁴/4, NOT x⁴. Students bring the exponent up (correct) but forget to divide (incorrect). Always write the fraction: xⁿ⁺¹/(n+1).
Try our integral calculator to verify any integral with step-by-step solutions.
Related: Derivative Calculator · Quadratic Formula Calculator · Square Root Calculator
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