∫e^(x^5 +8x^2 ) dx =((√π)/(4(√2)))e^x^5 erfi(2(√2)x)−((5(√π))/(4(128)(√2)))(super−erf_((hyper)) (2(√2)x))+c where[super−erf_((hyper)) (t)] is super−function in D_2 and [D


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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

	$i {d o n}^{'} t t h i n k w e c a n s o l v e t h i s w i t h o u t$ $u s i n g n e w s p e c i a l f u n c t i o n s l i k e t h i s o n e$ $i t r i e d w i t b h y p e r g e o m e t r i c f u n c t i o n$ $a n d m e j i e r G - f u n c t i o n (s p e c i a l h i g h f u n c t i o n)$ $b u t i {d i d n}^{'} t f i n d a n y s o l u t i o n s .$ $a n d i p o s t e d t h i s q u e s t i o n b e c a u s e m y b e$ $y o u h a v e a n i d e a t o s o l v e b e c a u s e y o u$ $a l w a y s d o .$ $i r e s p e c t y o u r m i n d s i r$ $t o p r o f . m a t h m i n d$
	Answered by M±th+et+s last updated on 12/Jun/20
	$I=∫e^(x^5 +8x^2 ) dx ∫e^x^5 e^(8x^2 ) dx { ((u(x)=e^x^5 du=5x^4 e^x^5 )),((dv=e^(8x^2 ) v=(1/4)(√(π/2))erfi(2(√2)x))) :} where erfi(y) is the imaginary eror function I=((√π)/(4(√2))) e^x^5 erfi(2(√2)x)−((5π)/(4(√2)))∫x^4 e^x^5 erfi(2(√2)x)dx...✠ I_1 =∫x^4 e^x^5 erfi(2(√2)x)dx;let 2(√2)x=u dx=(1/(2(√2)))dy I_1 =(1/(128(√2)))∫y^4 e^((1/(128(√2)))y^5 ) erfi(y)dy=r∫y^4 e^(ry^5 ) erfi(y) dy ;(r=(1/(128(√2)))∈R) but erfi(y)=(i/(√π))Σ_(k=0) ^∞ (((−1)^(k−1) .H_(2k+i) (iy))/(2^(3k+(1/2)) .k!(2k+1)))and J=∫y^n .e^(ry) .erfi(y)dy J=((−n!)/(√π))(−r)^(−(n+1)) exp(((−r^2 )/4)).Σ_(m=0) ^n (((−r)^m )/(m!)).Σ_(k=0) ^m ((m),(k) )(((−r)/2))^(m−k) ((r/2)+y)^(k+1) .[(−((r/2)+y)^2 +y)^2 ]^2 .Γ(((k+1)/2);−((r/2)+y)^2 ] −r^(−(n+1)) .erfi(y)Γ(A+1,ny) n∈N so ⇒⇒Γ(n+1,ry);n∈N super integrative conversion T^(super−T^(−1) ) [∫y^4 e^(ry^5 ) erfi(y)dy]=d(super−erf_(hyper) (y)) then I_1 =r∫y^4 e^(ry^5 ) erfi(y)dy=r∫d(super−erf_(hyper) (y)) I_1 =r super−erf_((hyper)) (y)+c=(1/(128(√2)))super−erf_((hyper)) (2(√2)x)+c so I=((√π)/(4(√2)))e^x^5 erfi(2(√2)x)−((5(√π))/(4(128)(√2)))[super−erf_(hyper) (2(√2)x)]+c$
	$I = \int e^{x^{5} + 8 x^{2}} d x$ $\int e^{x^{5}} e^{8 x^{2}} d x {\begin{cases} u (x) = e^{x^{5}} d u = 5 x^{4} e^{x^{5}} \\ d v = e^{8 x^{2}} v = \frac{1}{4} \sqrt{\frac{π}{2}} e r f i (2 \sqrt{2} x) \end{cases}$ $w h e r e e r f i (y) i s t h e i m a g i n a r y e r o r f u n c t i o n$ $I = \frac{\sqrt{π}}{4 \sqrt{2}} e^{x^{5}} e r f i (2 \sqrt{2} x) - \frac{5 π}{4 \sqrt{2}} \int x^{4} e^{x^{5}} e r f i (2 \sqrt{2} x) d x . . . ✠$ $I_{1} = \int x^{4} e^{x^{5}} e r f i (2 \sqrt{2} x) d x; l e t 2 \sqrt{2} x = u d x = \frac{1}{2 \sqrt{2}} d y$ $I_{1} = \frac{1}{128 \sqrt{2}} \int y^{4} e^{\frac{1}{128 \sqrt{2}} y^{5}} e r f i (y) d y = r \int y^{4} e^{{r y}^{5}} e r f i (y) d y; (r = \frac{1}{128 \sqrt{2}} \in R)$ $b u t e r f i (y) = \frac{i}{\sqrt{π}} \underset{k = 0}{\sum^{\infty}} \frac{{(- 1)}^{k - 1} . H_{2 k + i} (i y)}{2^{3 k + \frac{1}{2}} . k! (2 k + 1)} a n d J = \int y^{n} . e^{r y} . e r f i (y) d y$ $J = \frac{- n!}{\sqrt{π}} {(- r)}^{- (n + 1)} e x p (\frac{- r^{2}}{4}) . \underset{m = 0}{\sum^{n}} \frac{{(- r)}^{m}}{m!} . \underset{k = 0}{\sum^{m}} (\begin{matrix} m \\ k \end{matrix}) {(\frac{- r}{2})}^{m - k} {(\frac{r}{2} + y)}^{k + 1} . {[{(- {(\frac{r}{2} + y)}^{2} + y)}^{2}]}^{2} . Γ (\frac{k + 1}{2}; - {(\frac{r}{2} + y)}^{2}]$ $- r^{- (n + 1)} . e r f i (y) Γ (A + 1, n y) n \in N s o \Rightarrow\Rightarrow Γ (n + 1, r y); n \in N$ $s u p e r i n t e g r a t i v e c o n v e r s i o n \overset{s u p e r - T^{- 1}}{T} [\int y^{4} e^{{r y}^{5}} e r f i (y) d y] = d (s u p e r - {e r f}_{h y p e r} (y))$ $t h e n I_{1} = r \int y^{4} e^{{r y}^{5}} e r f i (y) d y = r \int d (s u p e r - {e r f}_{h y p e r} (y))$ $I_{1} = r s u p e r - {e r f}_{(h y p e r)} (y) + c = \frac{1}{128 \sqrt{2}} s u p e r - {e r f}_{(h y p e r)} (2 \sqrt{2} x) + c$ $s o$ $I = \frac{\sqrt{π}}{4 \sqrt{2}} e^{x^{5}} e r f i (2 \sqrt{2} x) - \frac{5 \sqrt{π}}{4 (128) \sqrt{2}} [s u p e r - {e r f}_{h y p e r} (2 \sqrt{2} x)] + c$
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