Integration

This lesson covers Integration as required by the A-Level Mathematics Pure 1 specification. Integration is the reverse process of differentiation. It is used to find areas under curves, recover functions from their derivatives, and solve a variety of mathematical problems. Integration and differentiation together form the core of calculus.

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Integration as the Reverse of Differentiation

If dy/dx = f(x), then y = ∫f(x) dx.

Integration "undoes" differentiation. Since differentiating a constant gives zero, integration always introduces an arbitrary constant C.

The Power Rule for Integration

If f(x) = xⁿ (where n ≠ −1), then:

∫xⁿ dx = xⁿ⁺¹ / (n + 1) + C

Key rule: "Add one to the power, divide by the new power, add C."

Function	Integral
xⁿ (n ≠ −1)	xⁿ⁺¹/(n + 1) + C
axⁿ	axⁿ⁺¹/(n + 1) + C
k (constant)	kx + C
x³	x⁴/4 + C
x⁻²	−x⁻¹ + C = −1/x + C
x^(1/2)	(2/3)x^(3/2) + C

Example: Find ∫(3x² − 4x + 5) dx.