Integration by Parts

When the integrand is a PRODUCT of two functions of $x$ that is not a simple substitution

pattern (i.e. no factor is the derivative of the argument of another), integration by parts

is the appropriate technique.

Derivation from the product rule. For differentiable functions $u(x)$ and $v(x)$, the

product rule gives $\dfrac{d}{dx}(uv)=u\dfrac{dv}{dx}+v\dfrac{du}{dx}$. Integrating both sides

with respect to $x$,

and rearranging gives the integration by parts formula:

In words: choose the integrand as a product $u\cdot(dv)$, where $dv$ is the PART that is easy to

integrate; the original integral is then replaced by a (hopefully simpler) new integral

$\int v\,du$.

Choosing $u$: the ILATE guide. Since the roles of $u$ and $dv$ are not interchangeable in

difficulty, a standard rule of thumb picks $u$ in the priority order Inverse trigonometric,

Logarithmic, Algebraic (polynomial), Trigonometric, Exponential -- whichever type

appears earliest in this list among the factors present is taken as $u$; the rest of the

integrand (together with $dx$) is $dv$. This ordering works because it tends to make $du$

simpler (a logarithm differentiates to an algebraic term) while $v$ (an antiderivative of $dv$)

stays manageable.

Worked illustration ($\int x\sin x\,dx$). Here $x$ is algebraic and $\sin x$ is

trigonometric, so by ILATE, $u=x$ (giving $du=dx$) and $dv=\sin x\,dx$ (giving $v=-\cos x$):

matching Example 4.

A single-function case: $\int\ln x\,dx$. When only ONE function is visible, write it as …