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Logarithmic integration

By the chain rule, we have

\begin{displaymath}
\frac{\mathrm{d}}{\mathrm{d}x} \ln{f(x)} = \frac{f'(x)}{f(x)}
\end{displaymath}

It therefore follows that

\begin{displaymath}
\int \frac{f'(x)}{f(x)} \mathrm{d}x = \ln{f(x)} + c
\end{displaymath}

This technique is known as logarithmic integration and can be used to evaluate integrals such as \( \int \frac{x \sin{x^2}}{\cos{x^2}}
\mathrm{d}x \), by noticing that \( \frac{\mathrm{d}}{\mathrm{d}x}
\cos{x^2} = 2 x \sin{x^2} \). Hence,

\begin{displaymath}
\int \frac{x \sin{x^2}}{\cos{x^2}} \mathrm{d}x = \frac{1}{2} \ln
\sin{x^2} + c
\end{displaymath}



Alexander Frolkin 2001-03-13