Substitution involves a few standard steps:

• Choose a suitable substitution, say u = f(x)
• Differentiate the substitution, to get \frac{du}{dx} = f'(x)
• Rearrange this derivative to get du = f'(x)dx
• Substitute both u = f(x) and du = f'(x)dx into the integral and see if you can simplify the result

Normally the necessary substitution will be given on the exam paper. Many functions that can be integrated by the substitution method can also be integrated by "inspection" or "observation". This means that they are in a form that fits a pattern. There are two common types:

"Top is derivative of bottom": \displaystyle \int \frac{f'(x)}{f(x)}dx = \ln[f(x)] + c

"A function times its derivative": \displaystyle \int f^n (x)f'(x)dx = \frac{1}{n+1}f^{n+1}(x) + c

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