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We may encounter the integral

$$\int\ln x\,\mathrm{d}x$$

and none of the techniques learnt so far seems to work. But here the derivative of a product of functions will be very useful:

$$(x\ln x)'=(x)'\ln x+x(\ln x)'=1\ln x+x\dfrac{1}{x}=\ln x+1$$

which is nearly what we need! Indeed

$$\int\ln x\,\mathrm{d}x=\int((x\ln x)'-1)\,\mathrm{d}x=x\ln x-x$$

Great! This technique is called integration by parts and works as follows: let $u(x)$ and $v(x)$ be some functions. We know that


so if our integrand is of the form $uv'$, we may switch to a completely different, and perhaps simpler, integral:

$$\int uv'\,\mathrm{d}x=\int((uv)'-u'v)\,\mathrm{d}x=uv-\int u'v\,\mathrm{d}x$$