The power rule for integration is valid for all values except when the exponent is equal to \(-1\). This is because \[\frac{d(\ln u)}{dx} = \frac{1}{u}\frac{du}{dx}\]
Thus, reversing the process where the denominator's exponent is \(-1\) would lead to an integral of the logarithmic form.
| \[\int\frac{1}{u}du=\ln |u|+C\] |
Thus, the integration of \[\int \frac{x \,dx}{3+x^2}\] will be done using the logarithmic form, whereas \[\int \frac{x\, dx}{(3+x^2)^2}\] will be done using the power rule for integration.
Example 1: Integrate the function \[\int_{1}^{2}\frac{1}{8-3x}dx\]
Solution: We can recognize this is an integral of logarithmic form because the denominator is to the power of -1 (e.g., it can be written as \((8-3x)^{-1}\).
Let \(u=8-3x\), \(du=-3dx\). We can substitute these values and change the variable to u
\[\frac{1}{-3}\int_{1}^{2}\frac{1}{u}du=\frac{1}{-3}\left[\ln |u|\right]_{1}^{2}\]
Solving for the definite integral we get,
\[\frac{\ln 2}{-3}-\frac{\ln 1}{-3} = \frac{\ln 2}{-3}\qquad \qquad (\ln 1=0)\]
Example 2: Integrate the function \[\int\frac{dx}{x(1+2\ln x)}\]
Solution: Let \(u=1+2\ln x\)
\begin{align} du &= \frac{2}{x}dx \\ \frac{dx}{x} &= \frac{du}{2} \end{align}
The integral after subitution is
\begin{align} &\int \frac{du}{2u} \\ &=\frac{\ln |u|}{2} +C \\ &=\frac{\ln |1+2\ln x|}{2} +C \end{align}
Example 3: Integrate \[\int \sec x\,dx\]
Solution: There is a trick here and that is to multiply \(\sec x\) by \[\frac{\sec x + \tan x}{\sec x + \tan x}\]
The resulting integral is
\[\int \frac{\sec ^2x + \sec x\tan x}{\sec x + \tan x}dx\]
Now let \(u=\sec x + \tan x\)
\[ du=\sec x\tan x + \sec ^2x \,dx\]
The integral becomes,
\[\int \frac{du}{u} = \ln |u| + C = \ln |\sec x + \tan x| + C \]
Designed by Matthew Cheung. This work is licensed under a Creative Commons Attribution 4.0 International License.