Sin-Inx-dx-

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} \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