Math Integration rules Integration by substitution

Integration by substitution

Like the chain rule when taking the derivative, integrating composite functions uses integration by substitution.



  1. Substitution: replace part of the function with $z$
  2. Adapt $\mathrm{d}x$ to $\mathrm{d}z$
  3. Integrate
  4. Substitute back

Converting the differential is done by the formula



$\int (3x+2)^3 \, \mathrm{d}x$

  1. Substitution

    We set $z$ in order to replace the difficult part.


    Insert $z$ in the function

    $\int (\color{red}{3x+2})^3 \, \mathrm{d}x$

    $\int \color{red}{z}^3 \, \mathrm{d}x$

  2. Adjust differential

    We change the differential with the formula:


    Take the derivative of $z$ for $z'$




    Convert for dx


  3. Integrate

    Insert the new differential into the integral.

    $\int z^3 \, \color{red}{\mathrm{d}x}$

    $\int z^3 \, \color{red}{\frac{\mathrm{d}z}{3}}$

    Rewrite the integral and integrate using known integration rules.

    $\int \frac13 z^3 \, \mathrm{d}z$ $=\frac1{12} z^4+C$

  4. Substitute back

    Now you are almost done. $z$ only has to be replaced again.


    $\frac1{12} \color{red}{z}^4+C$ $=\frac1{12}(\color{red}{3x+2})^4+C$

    So the solution is:

    $\int (3x+2)^3 \, \mathrm{d}x$ $=\frac1{12}(3x+2)^4+C$