Integration reverses differentiation. To integrate xⁿ (n ≠ −1), add one to the power and divide by the new power: ∫ xⁿ dx = xⁿ⁺¹/(n + 1) + c. So ∫ 6x² dx = 2x³ + c.
The constant c is needed because any constant disappears when you differentiate. Rewrite roots and fractions as powers first: ∫ 5/x² dx = ∫ 5x⁻² dx = −5x⁻¹ + c. If you are given a point on the curve, substitute it to find c.
Worked example
Find ∫ (6x + 2) dx.
- ∫ 6x dx = 3x² and ∫ 2 dx = 2x.
- Add the constant.
Answer:3x² + 2x + c
Key words
- integral
- the result of integrating
- constant of integration (c)
- an unknown constant added
- antiderivative
- a function whose derivative is the given one
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