Integration by parts

When the integrand is a product of functions of different nature (a polynomial times an exponential, a logarithm, etc.), one uses integration by parts, which arises from the product rule for differentiation read backwards.

Property — Integration by parts

$\int f'g\,dx = fg - \int fg'\,dx.$ Rule: $f'$ is integrated ($\to f$), $g$ is differentiated ($\to g'$). It must be chosen so that the “second piece” $\int fg'\,dx$ is simpler than the original.

Example — $\int_1^e\ln x\,dx = 1$

We choose $f'=1 \to f=x$; $g=\ln x \to g'=1/x$: $\bigl[x\ln x\bigr]_1^e - \int_1^e x\cdot\frac{1}{x}\,dx = (e\cdot 1 - 0) - [x]_1^e = e-(e-1) = \boxed{1}.$

Example — Wrong vs right choice: $\int xe^{2x}\,dx$

Wrong choice: $f'=x$, $g=e^{2x}$ $\to$ the second piece becomes $\int\dfrac{x^2}{2}\cdot 2e^{2x}\,dx$ — more complicated!

Right choice: $f'=e^{2x} \to f=\dfrac{e^{2x}}{2}$; $g=x \to g'=1$: $\frac{xe^{2x}}{2} - \int\frac{e^{2x}}{2}\,dx = \frac{xe^{2x}}{2} - \frac{e^{2x}}{4}+C.$

The practical rule: differentiate the factor that simplifies when differentiated (the polynomial, the logarithm), integrate the one that stays “equal to itself” (the exponential).

Links

Topics: Integrale
Concepts: Integrale per parti
Methods: Integrale parti
Skills: Calcolare · Integrare