User:Jounce/Integral list format

The following is a demo of aligning integral lists.

Indefinite integral
Indefinite integrals are antiderivative functions. A constant (the constant of integration) may be added to the right hand side of any of these formulas, but has been suppressed here in the interest of brevity.

Integrals involving polynomials (NEW VERSION)
$$ \begin{align}

\int xe^{cx}\; \mathrm{d}x &= e^{cx}\left(\frac{cx-1}{c^{2}}\right) \\

\int x^2 e^{cx}\;\mathrm{d}x &= e^{cx}\left(\frac{x^2}{c}-\frac{2x}{c^2}+\frac{2}{c^3}\right) \\

\int x^n e^{cx}\; \mathrm{d}x &= \frac{1}{c} x^n e^{cx} - \frac{n}{c}\int x^{n-1} e^{cx} \mathrm{d}x = \left( \frac{\partial}{\partial c} \right)^n \frac{e^{cx}}{c} = e^{cx}\sum_{i=0}^n (-1)^i\,\frac{n!}{(n-i)!\,c^{i+1}}\,x^{n-i} = e^{cx}\sum_{i=0}^n (-1)^{n-i}\,\frac{n!}{i!\,c^{n-i+1}}\,x^i \\

\int\frac{e^{cx}}{x}\; \mathrm{d}x &= \ln|x| +\sum_{n=1}^\infty\frac{(cx)^n}{n\cdot n!} \\

\int\frac{e^{cx}}{x^n}\; \mathrm{d}x &= \frac{1}{n-1}\left(-\frac{e^{cx}}{x^{n-1}}+c\int\frac{e^{cx} }{x^{n-1}}\,\mathrm{d}x\right) \qquad\mbox{(for }n\neq 1\mbox{)} \\ \end{align} $$

Integrals involving polynomials (OLD VERSION)

 * $$\int xe^{cx}\; \mathrm{d}x = e^{cx}\left(\frac{cx-1}{c^{2}}\right)$$


 * $$\int x^2 e^{cx}\;\mathrm{d}x = e^{cx}\left(\frac{x^2}{c}-\frac{2x}{c^2}+\frac{2}{c^3}\right)$$


 * $$\int x^n e^{cx}\; \mathrm{d}x = \frac{1}{c} x^n e^{cx} - \frac{n}{c}\int x^{n-1} e^{cx} \mathrm{d}x = \left( \frac{\partial}{\partial c} \right)^n \frac{e^{cx}}{c} = e^{cx}\sum_{i=0}^n (-1)^i\,\frac{n!}{(n-i)!\,c^{i+1}}\,x^{n-i} = e^{cx}\sum_{i=0}^n (-1)^{n-i}\,\frac{n!}{i!\,c^{n-i+1}}\,x^i$$


 * $$\int\frac{e^{cx}}{x}\; \mathrm{d}x = \ln|x| +\sum_{n=1}^\infty\frac{(cx)^n}{n\cdot n!}$$


 * $$\int\frac{e^{cx}}{x^n}\; \mathrm{d}x = \frac{1}{n-1}\left(-\frac{e^{cx}}{x^{n-1}}+c\int\frac{e^{cx} }{x^{n-1}}\,\mathrm{d}x\right) \qquad\mbox{(for }n\neq 1\mbox{)}$$