Example

Table 1: TABLE 1 Guidelines for Choosing \(u\) and \(dv\)

Integral; \(n\) is a positive integer	\(u\)	\(dv\)
\(\left. \begin{array}{l} \int x^{n}e^{ax}\,dx \\ \int x^{n}\cos (ax) \,dx \\ \int x^{n}\sin (ax) \,dx \end{array} \right\} \)	\(u=x^{n}\)	\(dv= \hbox{what remains}\)
\( \begin{array}{l} \int x^{n}\sin ^{-1}x\,dx \\ \int x^{n}\cos ^{-1}x\,dx \\ \int x^{n}\tan ^{-1}x\,dx \end{array} \)	\( \begin{array}{l} u=\sin ^{-1}x \\ u=\cos ^{-1}x \\ u=\tan ^{-1}x \end{array} \)	\(dv=x^{n}\,dx\)
\(\int x^{m}( \ln x) ^{n}\,dx;\) \(m\) is a real number, \(m ≠ -1\)	\(u=( \ln x) ^{n}\)	\(dv=x^{m}\,dx\)