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Mathematics LibreTexts

2.1: Substitution

  • Page ID
    521
  • [ "article:topic", "integration by substitution", "authorname:green" ]

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    Recall that the chain rule states that

    \[ (f(g(x)))' = f'(g(x))g'(x). \]

    Integrating both sides we get:

    \[ \int[f(g(x)]'dx = \int[f'(g(x)g'(x)dx]\]

    or

    \[ \int f'\left( g(x) \right) \, g' (x) \, dx = f\left(g(x)\right) + C \]

    Example 1

    Calculate

    \[ \int \dfrac{2x}{x^2+1}\, dx = \int 2x\left( x^2+1\right)^{-2} \, dx. \]

    Solution

    Let

    \[ u =  x^2 +1 \]

    then

    \[ \dfrac{du}{dx} = 2x \]

    and

    \[ du = 2x \,dx.\]

    We substitute:

     \[ \int u^{-2} du = -u^{-1} + C =  (x^2 +1)^{-1}  + C. \]

    Steps:

    1. Find the function derivative pair (\(f\) and \(f'\)).
    2. Let \(u = f(x)\).
    3. Find \(du/dx\) and adjust for constants.
    4. Substitute.
    5. Integrate.
    6. Resubstitute.

    We will try many more examples including those such as

     \[ \int x\, \sin(x^2)\, dx, \]

    \[ \int x\, \sqrt{x - 2}\, dx. \]

    Contributors