 Matematica
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Integrali con $\small e^{ax}$
- 1.
- $\int e^{ax} dx =\frac{e^{ax}}{a}$
- 2.
- $\displaystyle\int xe^{ax}dx=\displaystyle \frac{e^{ax}}{a}\left(x-\displaystyle \frac{1}{a}\right)$
- 3.
- $\displaystyle\int x^2 e^{ax}dx=\displaystyle \frac{e^{ax}}{a}\left(x^2-\displaystyle \frac{2x}{a}+\displaystyle \frac{2}{a^2}\right)$
- 4.
- $\begin{array}{lcl}
\displaystyle\int x^n e^{ax}dx&=& \displaystyle \frac{x^n e^{ax}}{a}-\displaystyle \frac{n}{a}\int x^{n-1}e^{ax}dx\\ &&\\ &=&\displaystyle \frac{e^{ax}}{a}\left(x^n-\displaystyle \frac{nx^{n-1}}{a}+\displaystyle \frac{n(n-1)x^{n-2}}{a^2}-\cdot\cdot\cdot \displaystyle \frac{(-1)^n n!}{a^n}\right)
\end{array}$
- 5.
- $\displaystyle\int\displaystyle \frac{e^{ax}}{x}dx=\ln x+\displaystyle \frac{ax}{1\cdot 1!}+\displaystyle \frac{(ax)^2}{2\cdot 2!}+\displaystyle \frac{(ax)^3}{3\cdot 3!}+\cdot\cdot\cdot$
- 6.
- $\displaystyle\int\displaystyle \frac{e^{ax}}{x^n}dx=\displaystyle \frac{-e^{ax}}{(n-1)x^{n-1}}+\displaystyle \frac{a}{n-1}\int\displaystyle \frac{e^{ax}}{x^{n-1}}dx$
- 7.
- $\displaystyle\int\displaystyle \frac{dx}{p+qe^{ax}}=\displaystyle \frac{x}{p}-\displaystyle \frac{1}{ap}\ln (p+qe^{ax})$
- 8.
- $\displaystyle\int\displaystyle \frac{dx}{(p+qe^{ax})^2}=\displaystyle \frac{x}{p ^2}+\displaystyle \frac{1}{ap(p+qe^{ax})}-\displaystyle \frac{1}{ap^2}\ln(p+qe^{ax})$
- 9.
- $\displaystyle\int\displaystyle \frac{dx}{pe^{ax}+qe^{-ax}}=\left\{ \begin{array}{ll}
\displaystyle \frac{1}{a\displaystyle \sqrt{pq}}\tan^{-1}\left(\displaystyle \sqrt{\displaystyle \frac{p}{q}}e^{ax}\right) \\\displaystyle \frac{1}{2a\displaystyle \sqrt{-pq}}\ln\left(\displaystyle \frac{e^{ax}-\displaystyle \sqrt{-q/p}}{e^{ax}+\displaystyle \sqrt{-q/p}}\right)
\end{array}
\right.$
- 10.
- $\displaystyle\int e^{ax}\sin bx dx=\displaystyle \frac{e^{ax}(a\sin bx -b\cos bx)}{a^2+b^2}$
- 11.
- $\displaystyle\int e^{ax}\cos bx dx=e^{ax}\displaystyle \frac{(a\cos bx+b\sin bx)}{a^2+b^2}$
- 12.
- $\small \displaystyle\int xe^{ax}\sin bx dx=\displaystyle \frac{xe^{ax}(a\sin bx -b\cos bx)}{a^2+b^2}-\displaystyle \frac{e^{ax}\left\{(a^2-b^2)\sin bx-2ab\cos bx\right\}}{(a^2+b^2)^2}$
- 13.
- $\small \displaystyle\int xe^{ax}\cos bx dx=\displaystyle \frac{xe^{ax}(a\cos bx +b\sin bx)}{a^2+b^2}-\displaystyle \frac{e^{ax}\left\{(a^2-b^2)\cos bx+2ab\sin bx\right\}}{(a^2+b^2)^2}$
- 14.
- $\displaystyle\int e^{ax}\ln xdx=\displaystyle \frac{e^{ax}\ln x}{a}-\displaystyle \frac{1}{a}\int\displaystyle \frac{e^{ax}}{x}dx$
- 15.
- $\small \displaystyle\int e^{ax}\sin^n bxdx=\displaystyle \frac{e^{ax}\sin^{n-1}bx}{a^2+n^2b^2}(a\sin bx-nb\cos bx) + \displaystyle \frac{n(n-1)b^2}{a^2+n^2b^2}\int e^{ax}\sin^{n-2}bx dx$
- 16.
- $\small \displaystyle\int e^{ax}\cos^n bxdx=\displaystyle \frac{e^{ax}\cos^{n-1}bx}{a^2+n^2b^2}(a\cos bx+nb\sin bx) + \displaystyle \frac{n(n-1)b^2}{a^2+n^2b^2}\int e^{ax}\cos^{n-2}bx dx$
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