This shows you the differences between two versions of the page.

Link to this comparison view

Both sides previous revision Previous revision
Next revision
Previous revision
function_integral_on_ℝⁿ [2016/03/28 20:22]
function_integral_on_ℝⁿ [2017/01/05 00:00]
Line 27: Line 27:
 Special case Special case
-$$\int_{-a}^a E(x) \left( 1 + \sum_{k=0}^\infty c_k U_k(x)^{2k+1} \right) = \int_0^a E(x) \,{\mathrm d}x$$+$$\int_{-a}^a E(x) \left( ​\dfrac{1}{2} + \sum_{k=0}^\infty c_k U_k(x)^{2k+1} \right) ​\,{\mathrm d}x = \int_0^a E(x) \,{\mathrm d}x$$
 e.g. all $U_k$ the same and $c_k$ so that you get $\frac{1}{1\pm e^{y}}$: e.g. all $U_k$ the same and $c_k$ so that you get $\frac{1}{1\pm e^{y}}$:
 $$\int_{-a}^a E(x) \dfrac{1}{1\pm {\mathrm e}^{U(x)}}\,​{\mathrm d}x = \int_0^a E(x) \,{\mathrm d}x$$ $$\int_{-a}^a E(x) \dfrac{1}{1\pm {\mathrm e}^{U(x)}}\,​{\mathrm d}x = \int_0^a E(x) \,{\mathrm d}x$$
 +$$\int_{-a}^a f(x^2) \dfrac{1}{1 + {\mathrm e}^{x^2\sin(x)}}\,​{\mathrm d}x = \int_0^a f(x^2) \,{\mathrm d}x$$
 === References === === References ===
Link to graph
Log In
Improvements of the human condition