Skip to main content
Mathematics LibreTexts

2.1: Substitution

  • Page ID
    521
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    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 and Attributions


    This page titled 2.1: Substitution is shared under a not declared license and was authored, remixed, and/or curated by Larry Green.

    • Was this article helpful?