6.3 · Integration by parts and partial fractions

Reverse the product rule

Integration by parts follows from integrating the product rule: $\displaystyle \int u\,dv=uv-\int v\,du$. Choose $\displaystyle u$ so that differentiating simplifies it, and choose $\displaystyle dv$ so that it can be integrated.

For $\displaystyle \int xe^x\,dx$, choose $\displaystyle u=x$ and $\displaystyle dv=e^x dx$. The remaining integral is simpler: $\displaystyle xe^x-\int e^x dx$. These techniques extend the AB core.

$$\int u\,dv=uv-\int v\,du$$

A logarithm can be the factor u

For $\displaystyle \int\ln x\,dx$ on $\displaystyle x>0$, think of the integrand as $\displaystyle \ln x\cdot1$. Choose $\displaystyle u=\ln x$, $\displaystyle dv=dx$, so $\displaystyle du=dx/x$ and $\displaystyle v=x$. The result is $\displaystyle x\ln x-x+C$.

If the remaining integral becomes harder, reconsider the choice of factors. Some integrals require repeated parts, but every stage must preserve the minus sign.

Break a rational function into simpler fractions

For distinct linear factors, express a proper rational function as a sum with unknown constants, clear denominators, and solve for those constants. If the numerator degree is at least the denominator degree, divide polynomials first.

For $\displaystyle 1/((x+1)(x+2))$, the decomposition is $\displaystyle 1/(x+1)-1/(x+2)$. Integrating gives logarithms with absolute values on intervals that avoid the denominator zeros.