∫e^(sin(x)) dx


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Question Number 96342 by Ar Brandon last updated on 31/May/20

$\int e^{\sin (x)} d x$
Commented by prakash jain last updated on 31/May/20

$You should clarify dx notation .$ $Are you asking :$ $\int e^{\sin x} d x ?$
Commented by Ar Brandon last updated on 31/May/20
OK
	Answered by mr W last updated on 31/May/20

	${what}^{'} s THE reason why you use this$ $FONT ?$
	Commented by Ar Brandon last updated on 31/May/20
	I just felt like doing so today. Any problem with that ? What's your opinion ?��
	Commented by prakash jain last updated on 31/May/20

	$Mathfrak font have special$ $meaning in algebra . So better$ $to follow convention especially$ $when typing math .$
	Commented by mr W last updated on 01/Jun/20

	${i t}^{'} s m a k e s t h e r e a d i n g u n n e c e s s a r l y$ $h a r d f o r o t h e r p e o p l e .$
	Commented by Ar Brandon last updated on 01/Jun/20
	Okay, thanks for your opinions. I'll make sure I use the usual font hencefort.��
	Answered by M±th+et+s last updated on 31/May/20
	$I=∫e^(sin(x)) dx e^u =Σ_(n=0) ^∞ (u^n /(n! )) , u=sin(x) ⇒e^(sin(x)) =Σ_(n=0) ^∞ (((sin(x))^n )/(n!)) I=∫e^(sin(x)) dx=∫Σ_(n=0) ^∞ (((sin(x))^n )/(n!))dx=Σ_(n=0) ^∞ (1/(n!))∫(sin(x))^n dx A=∫(sin(x))^n dx=−cos(x)∗(sin(x))^(n+1) ∗[(sin(x))^2 ]^(((−n)/2)−(1/2)) 2F1[(1/2),((1−n)/2);(3/2);(cos(x))^2 ]+c I=∫e^(sin(x)) dx=−Σ_(n=0) ^∞ (1/(n!))[A]+c notice\ 2F1 is special function called hyper geometric function$
	$I = \int e^{s i n (x)} d x$ $e^{u} = \underset{n = 0}{\sum^{\infty}} \frac{u^{n}}{n!}, u = s i n (x) \Rightarrow e^{s i n (x)} = \underset{n = 0}{\sum^{\infty}} \frac{{(s i n (x))}^{n}}{n!}$ $I = \int e^{s i n (x)} d x = \int \underset{n = 0}{\sum^{\infty}} \frac{{(s i n (x))}^{n}}{n!} d x = \underset{n = 0}{\sum^{\infty}} \frac{1}{n!} \int {(s i n (x))}^{n} d x$ $A = \int {(s i n (x))}^{n} d x = - c o s (x) * {(s i n (x))}^{n + 1} * {[{(s i n (x))}^{2}]}^{\frac{- n}{2} - \frac{1}{2}} 2 F 1 [\frac{1}{2}, \frac{1 - n}{2}; \frac{3}{2}; {(c o s (x))}^{2}] + c$ $I = \int e^{s i n (x)} d x = - \underset{n = 0}{\sum^{\infty}} \frac{1}{n!} [A] + c$ $n o t i c e ∖ 2 F 1 i s s p e c i a l f u n c t i o n c a l l e d$ $h y p e r g e o m e t r i c f u n c t i o n$
	Commented by Ar Brandon last updated on 31/May/20
	Thanks
	Commented by M±th+et+s last updated on 31/May/20

	$y o u a r e w e l c o m e$
	Commented by selea last updated on 01/Jun/20

	$sir can you help me ?$
	Commented by M±th+et+s last updated on 01/Jun/20

	$p o s t y o u r q u e s t i o n s s u r e i f i c a n i w i l l$
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