the electron wave in an atom is of type
Answers
Answer:
\textbf{Given:}Given:
\mathsf{\int\,sec^nu\;du}∫sec
n
udu
\textbf{To find:}To find:
\textsf{Reduction formula for}\;\mathsf{\int\,sec^nu\;du}Reduction formula for∫sec
n
udu
\textbf{Solution:}Solution:
\textsf{Consider,}Consider,
\mathsf{\int\,sec^nu\;du}∫sec
n
udu
\textsf{We apply Integration by parts formula}We apply Integration by parts formula
\boxed{\mathsf{\int\,m\,dn=mn-\int\,n\;dm}}
∫mdn=mn−∫ndm
\mathsf{=\int\,sec^{n-2}u\,(sec^2u\,du)}=∫sec
n−2
u(sec
2
udu)
\mathsf{Take,}Take,
\mathsf{m=sec^{n-2}u\;\implies\;dm=(n-2)\,sec^{n-3}u\;du}m=sec
n−2
u⟹dm=(n−2)sec
n−3
udu
\mathsf{dn=sec^u\,du\;\implies\;\int\,dn=\int\,sec^2u\,du\;\implies\;n=tanu}dn=sec
u
du⟹∫dn=∫sec
2
udu⟹n=tanu
\mathsf{Now,}Now,
\mathsf{\int\,sec^nu\;du=sec^{n-2}u\,tanu-\int\,tanu\,(n-2)sec^{n-3}u\,(secu\,tanu)du}∫sec
n
udu=sec
n−2
utanu−∫tanu(n−2)sec
n−3
u(secutanu)du
\mathsf{\int\,sec^nu\;du=sec^{n-2}u\,tanu-\int\,tanu\,(n-2)sec^{n-3}u(secu\,tanu)du}∫sec
n
udu=sec
n−2
utanu−∫tanu(n−2)sec
n−3
u(secutanu)du
\mathsf{\int\,sec^nu\;du=sec^{n-2}u\,tanu-(n-2)\int\,tan^2u\,sec^{n-2}u\,du}∫sec
n
udu=sec
n−2
utanu−(n−2)∫tan
2
usec
n−2
udu
\mathsf{\int\,sec^nu\;du=sec^{n-2}u\,tanu-(n-2)\int(sec^2u-1)\,sec^{n-2}u\,du}∫sec
n
udu=sec
n−2
utanu−(n−2)∫(sec
2
u−1)sec
n−2
udu
\mathsf{\int\,sec^nu\;du=sec^{n-2}u\,tanu-(n-2)\int(sec^nu-sec^{n-2}u)du}∫sec
n
udu=sec
n−2
utanu−(n−2)∫(sec
n
u−sec
n−2
u)du
\mathsf{\int\,sec^nu\;du=sec^{n-2}u\,tanu-(n-2)\int\,sec^nu\,du+(n-2)\int\,sec^{n-2}udu}∫sec
n
udu=sec
n−2
utanu−(n−2)∫sec
n
udu+(n−2)∫sec
n−2
udu
\mathsf{\int\,sec^nu\;du+(n-2)\int\,sec^nu\,du=sec^{n-2}u\,tanu+(n-2)\int\,sec^{n-2}udu}∫sec
n
udu+(n−2)∫sec
n
udu=sec
n−2
utanu+(n−2)∫sec
n−2
udu
\mathsf{(n-1)\int\,sec^nu\,du=sec^{n-2}u\,tanu+(n-2)\int\,sec^{n-2}udu}(n−1)∫sec
n
udu=sec
n−2
utanu+(n−2)∫sec
n−2
udu
\implies\boxed{\mathsf{\int\,sec^nu\,du=\dfrac{sec^{n-2}u\,tanu}{n-1}+\dfrac{n-2}{n-1}\int\,sec^{n-2}udu}}⟹
∫sec
n
udu=
n−1
sec
n−2
utanu
+
n−1
n−2
∫sec
n−2
udu
\textbf{Find more:}Find more:
Answer:
Here's the ans hope it's helpful for you☺
Explanation:
Along with all other quantum objects, an electron is partly a wave and partly a particle. To be more accurate, an electron is neither literally a traditional wave nor a traditional particle, but is instead a quantized fluctuating probability wavefunction.
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