Math ↠ Integration rules ↠ Integration by linear substitution

Integration by linear substitution

When integrating composite functions of the form $f(g(x))$ with a linear inner function, one uses the integration by linear substitution:

$\int f(mx+n) \, \mathrm{d}x$ $=\frac1m F(mx+n)+C$
!

Remember

Integration by linear substitution may only be used if the inner function $g(x)$ is a linear function (i.e.: $g(x)=mx+n$).

$f(g(x))$ $=f(mx+n)$
i

Hint

In addition to the integration by linear substitution, there is the integration by nonlinear substitution for arbitrarily composite functions.

Linear substitution is actually only a special case of general substitution, but it is sufficient for most tasks.

Examples

$\int (2x+4)^2 \, \mathrm{d}x$

  1. Split function into subfunctions

    $f(x)=x^2$ and $g(x)=\color{red}{2}x+4$

    $\color{red}{m=2}$

  2. Integrate $f(x)$

    Apply the power rule
    $F(x)=\color{blue}{\frac13}x\color{blue}{^3}$

  3. Insert

    $\int f(mx+n) \, \mathrm{d}x$ $=\frac{1}{\color{red}{m}} \cdot\color{blue}{F}(mx+n)+C$

    $\int (2x+4)^2 \, \mathrm{d}x$ $=\frac{1}{\color{red}{2}} \cdot \color{blue}{\frac13}(2x+4)^\color{blue}{3}+C$ $=\frac16(2x+4)^3+C$

$\int e^{2x} \, \mathrm{d}x$

  1. Split function into subfunctions

    $f(x)=e^x$ and $g(x)=\color{red}{2}x$

    $\color{red}{m=2}$

  2. Integrate $f(x)$

    Apply the power rule
    $F(x)=\color{blue}{e}^x$

  3. Insert

    $\int f(mx+n) \, \mathrm{d}x$ $=\frac{1}{\color{red}{m}} \cdot\color{blue}{F}(mx+n)+C$

    $\int e^{2x} \, \mathrm{d}x$ $=\frac{1}{\color{red}{2}} \cdot \color{blue}{e}^{2x}+C$
i

Hint

For exponential functions (second example) there is a "trick" to solve the integral even faster:

The antiderivative of an exponential function with linear inner function is obtained by dividing by the derivative of the inner function.

$\int e^{g(x)} \, \mathrm{d}x=\frac{e^{g(x)}}{g'(x)}+C$
Example: $\int e^{2x} \, \mathrm{d}x=\frac{e^{2x}}{2}+C$