A.4. Integrali esponenziali e logaritmici — formule e risorse

\int u\,e^{au}\,du = \frac{1}{a^2}(au - 1)\,e^{au} + C

\int u^n e^{au}\,du = \frac{1}{a}\,u^n e^{au} - \frac{n}{a}\int u^{n-1} e^{au}\,du

\int e^{au}\,\sin(bu)\,du = \frac{e^{au}}{a^2 + b^2}\left(a\,\sin(bu) - b\,\cos(bu)\right) + C

\int e^{au}\,\cos(bu)\,du = \frac{e^{au}}{a^2 + b^2}\left(a\,\cos(bu) + b\,\sin(bu)\right) + C

\int \ln u\,du = u\,\ln u - u + C

\int \frac{du}{u\,\ln u} = \ln|\ln u| + C

\int u^n \ln u\,du = \frac{u^{n+1}}{(n+1)^2}\left[(n+1)\ln u - 1\right] + C

\int u^m \ln^n u\,du = \frac{u^{m+1}\ln^n u}{m+1} - \frac{n}{m+1}\int u^m \ln^{n-1} u\,du, \quad m \neq -1

\int \ln\left(u^2 + a^2\right)\,du = u\,\ln\left(u^2 + a^2\right) - 2u + 2a\,\arctan\frac{u}{a} + C

\int \ln\left|u^2 - a^2\right|\,du = u\,\ln\left|u^2 - a^2\right| - 2u + a\,\ln\left|\frac{u+a}{u-a}\right| + C

\int \frac{du}{a + be^u} = \frac{u}{a} - \frac{1}{a}\,\ln\left|a + be^u\right| + C