Integration by substitution

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

Converting the differential is done by the formula

Example

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

1. Substitution

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

   $z=3x+2$

   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:

   $\frac{\mathrm{d}z}{\mathrm{d}x}=z'$

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

   $z'=(3x+2)'=3$

   Insert

   $\frac{\mathrm{d}z}{\mathrm{d}x}=3$

   Convert for dx

   $\mathrm{d}x=\frac{\mathrm{d}z}{3}$

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.

   $z=3x+2$

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

   So the solution is:

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