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Integration - Basics

Maths: Edexcel A Level Pure Maths, topic 8.1.1

Integration

Integration is the opposite of differentiation.

Integration

Integration

  • To integrate xnx^n, add one to the power of x and divide by the new power.
  • xn→xn+1n+1x^n \rightarrow \frac{x^{n+1}}{n+1}
  • Terms with no powers of xx differentiate to nothing.
  • So we must add an extra unknown "constant of integration" after we integrate:
  • xn→xn+1n+1+cx^n \rightarrow \frac{x^{n+1}}{n+1} + c
  • This rule only works for n≠1n \ne 1.
Example

Example

  • If y=x3−5y = x^3 - 5, then dydx=3x2\frac{dy}{dx}= 3x^2.
  • If y=x3+100y = x^3 +100, then dydx=3x2\frac{dy}{dx}= 3x^2.
  • So if we integrate dydx=3x2\frac{dy}{dx}= 3x^2, we have to add the extra term:
  • dydx=3x2→y=x3+c\frac{dy}{dx}= 3x^2 \rightarrow y = x^3 + c
Constant of integration

Constant of integration

  • To find the constant of integration, we need to know a value of yy for a specific value of xx.
  • Substitute these values into the equation after integrating and rearrange for cc.
Example

Example

  • What is the equation of the curve that has gradient dydx=x2+2\frac{dy}{dx} = x^2 + 2 and passes through the point (3,12)(3,12)?
  • The gradient of a curve is equal to the differential of the equation of the curve, so we must integrate the gradient to get the equation of the curve.
  • dydx=x2+2→y=x33+2x+c\frac{dy}{dx} = x^2 + 2 \rightarrow y=\frac{x^3}{3} + 2x + c
Example

Example

  • To find cc, substitute x=3x = 3 and y=12y = 12:
  • 12=333+6+c12 = \frac{3^3}{3} + 6 + c
  • c=−3c = -3
  • So the equation of the curve is y=x33+2x−3y = \frac{x^3}{3} + 2x-3

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