| 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, \(m ≠ -1\) | \(u=( \ln x) ^{n}\) | \(dv=x^{m}\,dx\) |