• MathsFirst Home
• Online Maths Help
  • Notation
  • Arithmetic
    • Decimals
    • Fractions
    • HCF and LCM
    • Order of Operations
    • Signed Numbers
  • Algebra
    • Exponents
      • Integer Exponents
      • Fractional Exponents
      • Exponents Worksheet
    • Combining Like Terms
    • Simple Expansions
    • Factorisation
    • Mathematical Formulae
      • Constructing Formulae
      • Rearranging Formulae
        • Order of Operations for Algebraic Expressions
        • Inverse Operations and Functions
        • Rearranging Equations I (Simple Equations)
        • Rearranging Equations II (Quadratic Equations)
        • Rearranging Equations III (Harder Examples)
      • Area, Surface Area and Volume Formulae
    • Coordinate Systems and Graphs
    • Linear Equations & Graphs
    • Simultaneous Linear Equations
    • Polynomials
    • Quadratic Polynomials
    • Rational Functions
    • Logarithms
  • Calculus
    • Differentiation
      • Tangents, Derivatives and Differentiation
      • The Basic Rules
      • Products and Quotients
      • Chain Rule
      • Logarithmic Differentiation
      • Mixed Differentiation Problems
      • Sign of the Derivative
      • Sign of the Second Derivative
    • Integration
      • Basic Integrals
      • Multiple, Sum and Difference Rules
      • Linear Substitution
      • Simpler Integration by Substitution
      • Harder Integration by Substitution
      • Trig Substitution 1
      • Trig Substitution 2
      • Integration by Parts
  • Trigonometry
    • Pythagoras Theorem
    • Trig Functions and Related Topics
• First Year Maths Pathways
• Readiness Quizzes
• Study Maths
  at Massey
  • Undergraduate
  • Postgraduate
• Other Math Links
• About MathsFirst
• Contact Us

Integration

Integration by Substitution 1

We assume that you are familiar with basic integration.

Recall that if , then the indefinite integral f(x) dx = F(x) + c.

Note that there are no general integration rules for products and quotients of two functions.

We now provide a rule that can be used to integrate products and quotients in particular forms. The idea is to convert an integral into a basic one by substitution.

Before we give a general expression, we look at an example.

Example

By the chain rule,

Hence cos(x²)(2x) dx = sin(x²) + c…… (#)

If we substitute u = x² then = 2x.

The left hand side of (#) becomes cos(u)() dx, and the right hand side is sin(u) + c which equals cos(u) du.

Thus, cos(u)() dx = cos(u) du , where u = x².

This can be generalised for any product that is a composite function multiplied by the derivative of the inner funcion.

This integration rule follows from the chain rule for differentiation.

Substitution Rule

To integrate f(g(x))g'(x) dx

1. Convert it to f(u) du by substituting u for g(x) and substituting du for g'(x) dx, taking out a common numeric factor, n, if present.
2. Evaluate f(u) du = F(u) + c or n f(u) du = nF(u) + c
3. Replace u by g(x) in F(u)

NB. The presence of the derivative as a factor of what is being substituted into an integrand is an essential ingredient of the substitution rule.

Examples

Evaluate

Evaluate

Evaluate

Exercises

The hardest part of the integration is often choosing the right value to substitute. These exercises give practice in choosing u.

These exercises are more tricky. Watch out for a common numeric factor in the derivative factor!

Now try solving the whole problem. Evaluate the indefinite integrals.

Quotients

One type of problem deserves special mention. The problem appears at first sight as a quotient with larger denominator than numerator rather than as a product of a function raised to the power -1 and its derivative and thus often fools solvers when viewed in isolation:

= f(x)^-1 f '(x) dx = ln | f(x) | + c

The derivative, f '(x), "disappears like magic".

Examples

Be very careful to check that the numerator is equal to a constant times the derivative of the denominator or this method cannot be used.

Example

The derivative of x² + x − 1 is 2x + 1 not just 2x so this method cannot be used.

Integration Index | Integration by Substitution 2 >>