∫Sin(Inx)dx


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Question Number 201546 by Calculusboy last updated on 08/Dec/23

$\int S i n (I n x) d x$
Commented by aleks041103 last updated on 09/Dec/23

$i s t h i s s i n e o f n a t u r a l l o g o f x o r s t h e l s e ?$
Commented by Calculusboy last updated on 09/Dec/23

$y e s i t i s s i n e o f n a t u r a l l o g o f x$
Commented by mr W last updated on 09/Dec/23

$w h y {d o n}^{'} t y o u m a k e i t c l e a r s i r$ $t o a v o i d c o n f u s i n g o t h e r p e o p l e$ $a g a i n a n d a g a i n ?$ $h e r e y o u m e a n I n (x) i s L n (x),$ $b u t i n Q 201229 y o u s a i d I n (x) i s$ $n o t L n (x), b u t s o m e t h i n g e l s e .$ $a s i k n o w, n a t u r a l l o g a r i t h m i s$ $g e n e r a l l y w r i t t e n a s L n (x) o r$ ${L o g}_{e} (x) o r L o g (x), b u t n o t a s I n (x) .$ $s o p l e a s e w r i t e l n (x) i f y o u m e a n$ $n a t u r a l l i g a r i t h m, n o t a s I n (x) .$
Commented by Calculusboy last updated on 09/Dec/23

$o k a y, a m s o r r y f o r c o n f u s i n g$
	Answered by Mathspace last updated on 09/Dec/23
	$x=e^t ⇒I=∫ sin(t)e^t dt =Im(∫ e^(it+t) dt) and ∫ e^((1+i)t) dt=(1/(1+i))e^((1+i)t) =((1−i)/2)e^t {cost+isint} =(e^t /2){cost+isint−icost +sint} ⇒I=(e^t /2){sint−cost} +λ =(x/2){sin(lnx)−cos(lnx)}+λ$
	$x = e^{t} \Rightarrow I = \int s i n (t) e^{t} d t$ $= I m (\int e^{i t + t} d t)$ $a n d \int e^{(1 + i) t} d t = \frac{1}{1 + i} e^{(1 + i) t}$ $= \frac{1 - i}{2} e^{t} {c o s t + i s i n t}$ $= \frac{e^{t}}{2} {c o s t + i s i n t - i c o s t + s i n t}$ $\Rightarrow I = \frac{e^{t}}{2} {s i n t - c o s t} + λ$ $= \frac{x}{2} {s i n (l n x) - c o s (l n x)} + λ$
	Answered by JDamian last updated on 09/Dec/23

	$C a l c u l u s b o y d e f i n i t e l y c a n n o t d i f f e r e n t i a t e$ $a l o w e r c a s e L f r o m a n u p p e r c a s e I .$
	Commented by Calculusboy last updated on 09/Dec/23

	$i k n o w t h e d i f f e r e n c e, i j u s t m i s t y p e i t w h e n i$ $w a n t t o p o s t i t$
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