Home/Calculus II/Appendix A: extensive table of antiderivatives/A.7. Integrals with a2+u2\sqrt{a^2 + u^2}a2+u2ContentA.7. Integrals with a2+u2\sqrt{a^2 + u^2}a2+u2Usual real conditions: a>0a>0a>0; when uuu appears in the denominator, u≠0u\neq0u=0.FormulaCopy LaTeX∫a2+u2 du=u2 a2+u2+a22 ln∣u+a2+u2∣+C\int \sqrt{a^2 + u^2}\,du = \frac{u}{2}\,\sqrt{a^2 + u^2} + \frac{a^2}{2}\,\ln\left|u + \sqrt{a^2 + u^2}\right| + C∫a2+u2du=2ua2+u2+2a2lnu+a2+u2+Cln∣u+a2+u2∣+C\ln\left|u + \sqrt{a^2 + u^2}\right| + Clnu+a2+u2+CView detail →FormulaCopy LaTeX∫u2 a2+u2 du=u8(a2+2u2)a2+u2−a48 ln∣u+a2+u2∣+C\int u^2\,\sqrt{a^2 + u^2}\,du = \frac{u}{8}\left(a^2 + 2u^2\right)\sqrt{a^2 + u^2} - \frac{a^4}{8}\,\ln\left|u + \sqrt{a^2 + u^2}\right| + C∫u2a2+u2du=8u(a2+2u2)a2+u2−8a4lnu+a2+u2+Cln∣u+a2+u2∣+C\ln\left|u + \sqrt{a^2 + u^2}\right| + Clnu+a2+u2+CView detail →FormulaCopy LaTeX∫a2+u2u du=a2+u2−a ln∣a+a2+u2u∣+C\int \frac{\sqrt{a^2 + u^2}}{u}\,du = \sqrt{a^2 + u^2} - a\,\ln\left|\frac{a + \sqrt{a^2 + u^2}}{u}\right| + C∫ua2+u2du=a2+u2−alnua+a2+u2+Cln∣a+a2+u2u∣+C\ln\left|\frac{a + \sqrt{a^2 + u^2}}{u}\right| + Clnua+a2+u2+CView detail →FormulaCopy LaTeX∫a2+u2u2 du=−a2+u2u+ln∣u+a2+u2∣+C\int \frac{\sqrt{a^2 + u^2}}{u^2}\,du = -\frac{\sqrt{a^2 + u^2}}{u} + \ln\left|u + \sqrt{a^2 + u^2}\right| + C∫u2a2+u2du=−ua2+u2+lnu+a2+u2+CView detail →FormulaCopy LaTeX∫dua2+u2=ln∣u+a2+u2∣+C\int \frac{du}{\sqrt{a^2 + u^2}} = \ln\left|u + \sqrt{a^2 + u^2}\right| + C∫a2+u2du=lnu+a2+u2+CView detail →FormulaCopy LaTeX∫u2 dua2+u2=u2 a2+u2−a22 ln∣u+a2+u2∣+C\int \frac{u^2\,du}{\sqrt{a^2 + u^2}} = \frac{u}{2}\,\sqrt{a^2 + u^2} - \frac{a^2}{2}\,\ln\left|u + \sqrt{a^2 + u^2}\right| + C∫a2+u2u2du=2ua2+u2−2a2lnu+a2+u2+Cln∣u+a2+u2∣+C\ln\left|u + \sqrt{a^2 + u^2}\right| + Clnu+a2+u2+CView detail →FormulaCopy LaTeX∫duu a2+u2=−1a ln∣a2+u2+au∣+C\int \frac{du}{u\,\sqrt{a^2 + u^2}} = -\frac{1}{a}\,\ln\left|\frac{\sqrt{a^2 + u^2} + a}{u}\right| + C∫ua2+u2du=−a1lnua2+u2+a+Cln∣a2+u2+au∣+C\ln\left|\frac{\sqrt{a^2 + u^2} + a}{u}\right| + Clnua2+u2+a+CView detail →FormulaCopy LaTeX∫duu2 a2+u2=−a2+u2a2u+C\int \frac{du}{u^2\,\sqrt{a^2 + u^2}} = -\frac{\sqrt{a^2 + u^2}}{a^2 u} + C∫u2a2+u2du=−a2ua2+u2+CView detail →FormulaCopy LaTeX∫du(a2+u2)3/2=ua2 a2+u2+C\int \frac{du}{\left(a^2 + u^2\right)^{3/2}} = \frac{u}{a^2\,\sqrt{a^2 + u^2}} + C∫(a2+u2)3/2du=a2a2+u2u+CView detail →
FormulaCopy LaTeX∫a2+u2 du=u2 a2+u2+a22 ln∣u+a2+u2∣+C\int \sqrt{a^2 + u^2}\,du = \frac{u}{2}\,\sqrt{a^2 + u^2} + \frac{a^2}{2}\,\ln\left|u + \sqrt{a^2 + u^2}\right| + C∫a2+u2du=2ua2+u2+2a2lnu+a2+u2+Cln∣u+a2+u2∣+C\ln\left|u + \sqrt{a^2 + u^2}\right| + Clnu+a2+u2+CView detail →
FormulaCopy LaTeX∫u2 a2+u2 du=u8(a2+2u2)a2+u2−a48 ln∣u+a2+u2∣+C\int u^2\,\sqrt{a^2 + u^2}\,du = \frac{u}{8}\left(a^2 + 2u^2\right)\sqrt{a^2 + u^2} - \frac{a^4}{8}\,\ln\left|u + \sqrt{a^2 + u^2}\right| + C∫u2a2+u2du=8u(a2+2u2)a2+u2−8a4lnu+a2+u2+Cln∣u+a2+u2∣+C\ln\left|u + \sqrt{a^2 + u^2}\right| + Clnu+a2+u2+CView detail →
FormulaCopy LaTeX∫a2+u2u du=a2+u2−a ln∣a+a2+u2u∣+C\int \frac{\sqrt{a^2 + u^2}}{u}\,du = \sqrt{a^2 + u^2} - a\,\ln\left|\frac{a + \sqrt{a^2 + u^2}}{u}\right| + C∫ua2+u2du=a2+u2−alnua+a2+u2+Cln∣a+a2+u2u∣+C\ln\left|\frac{a + \sqrt{a^2 + u^2}}{u}\right| + Clnua+a2+u2+CView detail →
FormulaCopy LaTeX∫a2+u2u2 du=−a2+u2u+ln∣u+a2+u2∣+C\int \frac{\sqrt{a^2 + u^2}}{u^2}\,du = -\frac{\sqrt{a^2 + u^2}}{u} + \ln\left|u + \sqrt{a^2 + u^2}\right| + C∫u2a2+u2du=−ua2+u2+lnu+a2+u2+CView detail →
FormulaCopy LaTeX∫dua2+u2=ln∣u+a2+u2∣+C\int \frac{du}{\sqrt{a^2 + u^2}} = \ln\left|u + \sqrt{a^2 + u^2}\right| + C∫a2+u2du=lnu+a2+u2+CView detail →
FormulaCopy LaTeX∫u2 dua2+u2=u2 a2+u2−a22 ln∣u+a2+u2∣+C\int \frac{u^2\,du}{\sqrt{a^2 + u^2}} = \frac{u}{2}\,\sqrt{a^2 + u^2} - \frac{a^2}{2}\,\ln\left|u + \sqrt{a^2 + u^2}\right| + C∫a2+u2u2du=2ua2+u2−2a2lnu+a2+u2+Cln∣u+a2+u2∣+C\ln\left|u + \sqrt{a^2 + u^2}\right| + Clnu+a2+u2+CView detail →
FormulaCopy LaTeX∫duu a2+u2=−1a ln∣a2+u2+au∣+C\int \frac{du}{u\,\sqrt{a^2 + u^2}} = -\frac{1}{a}\,\ln\left|\frac{\sqrt{a^2 + u^2} + a}{u}\right| + C∫ua2+u2du=−a1lnua2+u2+a+Cln∣a2+u2+au∣+C\ln\left|\frac{\sqrt{a^2 + u^2} + a}{u}\right| + Clnua2+u2+a+CView detail →
FormulaCopy LaTeX∫duu2 a2+u2=−a2+u2a2u+C\int \frac{du}{u^2\,\sqrt{a^2 + u^2}} = -\frac{\sqrt{a^2 + u^2}}{a^2 u} + C∫u2a2+u2du=−a2ua2+u2+CView detail →
FormulaCopy LaTeX∫du(a2+u2)3/2=ua2 a2+u2+C\int \frac{du}{\left(a^2 + u^2\right)^{3/2}} = \frac{u}{a^2\,\sqrt{a^2 + u^2}} + C∫(a2+u2)3/2du=a2a2+u2u+CView detail →