Math Integration rules Integration by substitution

# Integration by substitution

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

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### Method

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

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

### 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

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

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

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:

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