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Integration · Topic 1 of 9

Introduction

Video lesson

Theory

We have previously learned how to differentiate terms.

If f(x)=xnf(x)=x^n then f′(x)=nxn−1f'(x)=nx^{n-1} (n∈Rn \in \mathbb{R}).

Say we are given the derivatives of some functions. Can we find the original functions?

  • f′(x)=9x2  ⟹  f(x)=3x3f'(x)=9x^2 \implies f(x) = 3x^3
  • f′(x)=−8x−3  ⟹  f(x)=4x−2f'(x)=-8x^{-3} \implies f(x) = 4x^{-2}
  • f′(x)=5  ⟹  f(x)=5xf'(x)=5 \implies f(x) = 5x
  • f′(x)=0  ⟹  f(x)=cf'(x)=0 \implies f(x) = c (where cc is a constant)

Integration is a branch of calculus. It is the inverse of differentiation.

⚠️ Common Examiner Traps

  • Integration undoes differentiation: every check you make should be "does differentiating my answer give me back the original?" It catches most errors in seconds.
  • The constant is part of the answer: an indefinite integral without + C+\,C is incomplete, and the working that follows it can be treated as invalid.
  • Notation is assessed: write the integral sign and the dxdx together. An integral with no dxdx is not a complete statement.
  • Rewrite before integrating: as with differentiation, every term needs to be in the form axnax^n first.