Logarithmic integration

By the chain rule, we have

$$\frac{\mathrm{d}}{\mathrm{d}x} \ln{f(x)} = \frac{f'(x)}{f(x)}$$

It therefore follows that

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

This technique is known as logarithmic integration and can be used to evaluate integrals such as , by noticing that . Hence,

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