e-x-5-8x-2-dx-pi-4-2-e-x-5-erfi-2-2-x-5-pi-4-128-2-super-erf-hyper-2-2-x-c-where-super-erf-hyper-t-is-super-function-in-D-2-and-D-n-

Question Number 98151 by M±th+et+s last updated on 11/Jun/20

\int e^{x^{5} + 8 x^{2}} d x

= \frac{\sqrt{π}}{4 \sqrt{2}} e^{x^{5}} e r f i (2 \sqrt{2} x) - \frac{5 \sqrt{π}}{4 (128) \sqrt{2}} (s u p e r - {e r f}_{(h y p e r)} (2 \sqrt{2} x)) + c

w h e r e [s u p e r - {e r f}_{(h y p e r)} (t)] i s s u p e r - f u n c t i o n

i n D_{2} a n d [D_{n}]

Answered by maths mind last updated on 12/Jun/20

i d i d n t f i n d t h i s s u p e r - {e r f}_{h y o e r} (t) c a n y o u e x p r e s s d e f i n i t i o n

Commented by M±th+et+s last updated on 12/Jun/20

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Answered by M±th+et+s last updated on 12/Jun/20

I = \int e^{x^{5} + 8 x^{2}} d x

\int e^{x^{5}} e^{8 x^{2}} d x {\begin{cases} u (x) = e^{x^{5}} d u = 5 x^{4} e^{x^{5}} \\ d v = e^{8 x^{2}} v = \frac{1}{4} \sqrt{\frac{π}{2}} e r f i (2 \sqrt{2} x) \end{cases}

w h e r e e r f i (y) i s t h e i m a g i n a r y e r o r f u n c t i o n

I = \frac{\sqrt{π}}{4 \sqrt{2}} e^{x^{5}} e r f i (2 \sqrt{2} x) - \frac{5 π}{4 \sqrt{2}} \int x^{4} e^{x^{5}} e r f i (2 \sqrt{2} x) d x \dots ✠

I_{1} = \int x^{4} e^{x^{5}} e r f i (2 \sqrt{2} x) d x; l e t 2 \sqrt{2} x = u d x = \frac{1}{2 \sqrt{2}} d y

I_{1} = \frac{1}{128 \sqrt{2}} \int y^{4} e^{\frac{1}{128 \sqrt{2}} y^{5}} e r f i (y) d y = r \int y^{4} e^{{r y}^{5}} e r f i (y) d y; (r = \frac{1}{128 \sqrt{2}} \in R)

b u t e r f i (y) = \frac{i}{\sqrt{π}} \underset{k = 0}{\sum^{\infty}} \frac{{(- 1)}^{k - 1} . H_{2 k + i} (i y)}{2^{3 k + \frac{1}{2}} . k! (2 k + 1)} a n d J = \int y^{n} . e^{r y} . e r f i (y) d y

J = \frac{- n!}{\sqrt{π}} {(- r)}^{- (n + 1)} e x p (\frac{- r^{2}}{4}) . \underset{m = 0}{\sum^{n}} \frac{{(- r)}^{m}}{m!} . \underset{k = 0}{\sum^{m}} (\begin{matrix} m \\ k \end{matrix}) {(\frac{- r}{2})}^{m - k} {(\frac{r}{2} + y)}^{k + 1} . {[{(- {(\frac{r}{2} + y)}^{2} + y)}^{2}]}^{2} . Γ (\frac{k + 1}{2}; - {(\frac{r}{2} + y)}^{2}]

- r^{- (n + 1)} . e r f i (y) Γ (A + 1, n y) n \in N s o \Rightarrow\Rightarrow Γ (n + 1, r y); n \in N

s u p e r i n t e g r a t i v e c o n v e r s i o n \overset{s u p e r - T^{- 1}}{T} [\int y^{4} e^{{r y}^{5}} e r f i (y) d y] = d (s u p e r - {e r f}_{h y p e r} (y))

t h e n I_{1} = r \int y^{4} e^{{r y}^{5}} e r f i (y) d y = r \int d (s u p e r - {e r f}_{h y p e r} (y))

I_{1} = r s u p e r - {e r f}_{(h y p e r)} (y) + c = \frac{1}{128 \sqrt{2}} s u p e r - {e r f}_{(h y p e r)} (2 \sqrt{2} x) + c

s o

I = \frac{\sqrt{π}}{4 \sqrt{2}} e^{x^{5}} e r f i (2 \sqrt{2} x) - \frac{5 \sqrt{π}}{4 (128) \sqrt{2}} [s u p e r - {e r f}_{h y p e r} (2 \sqrt{2} x)] + c