1-find-2-ln-x-dx-2-prove-i-1-i-resl-number-

Question Number 87833 by M±th+et£s last updated on 06/Apr/20

1) f i n d \int 2^{l n (x)} d x

2) p r o v e \sqrt[i]{i} = r e s l n u m b e r

Commented by mr W last updated on 06/Apr/20

2)

i = e^{\frac{i π}{2}}

\sqrt[i]{i} = i^{\frac{1}{i}} = {(e^{\frac{i π}{2}})}^{\frac{1}{i}} = e^{\frac{π}{2}} = \sqrt{e^{π}} = r e a l

Commented by john santu last updated on 06/Apr/20

1) by parts

u = 2^{\ln x} \Rightarrow du = 2^{\ln x} . \frac{\ln 2}{x} dx

I = x . 2^{\ln x} - \ln 2 \int 2^{\ln x} dx

(1 + \ln 2) I = x . 2^{\ln x}

I = \frac{x . 2^{\ln x}}{\ln (2 e)} + c

Commented by M±th+et£s last updated on 06/Apr/20

t h a n k y o u

Commented by M±th+et£s last updated on 06/Apr/20

t a n k y o u

Commented by mathmax by abdo last updated on 06/Apr/20

2)^{i} \sqrt{i} = i^{\frac{1}{i}} = i^{- i} = {(e^{\frac{i π}{2}})}^{- i} = e^{\frac{π}{2}} a n d e^{\frac{π}{2}} \in R

Answered by petrochengula last updated on 06/Apr/20

\int 2^{l n x} d x

e^{l n 2} = 2 t h e n 2^{l n x} = {(e^{l n 2})}^{l n x} = {(2^{l n x})}^{l n 2} = x^{l n 2}

\int 2^{l n x} d x = \int x^{l n 2} d x = \frac{x^{l n 2 + 1}}{l n 2 + 1} + C