Question

calculate integration of 5^5^x ×5^x dx​

Answer 1

5 power 5

so

5×5×5×5×5× 5×x

Answer 2

$\begin{gathered}\Large{\bold{\pink{\underline{Formula \: Used \::}}}} \end{gathered}$

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$1. \: \boxed{ \blue {\rm \: \dfrac{d}{dx} \: {a}^{x} = {a}^{x} log(a) }}$

$2. \: \boxed{ \blue {\rm \: \int \: {a}^{x} dx \: = \dfrac{ {a}^{x} }{ log(a) } + c}}$

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$\large\underline\purple{\bold{Solution :- }}$

$:\implies \tt \:Let \: I \: = \int \: {5}^{ {5}^{x} } {5}^{x} \: dx$

$\red{ \rm \: Put \: {5}^{x} = y}$

On differentiating both sides w. r. t. x,we get

$:\implies \tt \: \red{ \rm \: {5}^{x} log(5) = \dfrac{dy}{dx} }$

$:\implies \tt \: \red{ \rm \: {5}^{x} dx = \dfrac{dy}{ log(5) } }$

$:\implies \tt \: I \: = \int \: {5}^{y} \dfrac{dy}{ log(5) }$

$:\implies \tt \: I = \dfrac{1}{ log(5) } \int \: {5}^{y} dy$

$:\implies \tt \: I = \dfrac{1}{ log(5) } \times \dfrac{ {5}^{y} }{ log(5) } + c$

$:\implies \tt \: I = \dfrac{ {5}^{y} }{ {( log(5) )}^{2} } + c$

$:\implies \tt \: I = \dfrac{ {5}^{ {5}^{x} } }{ {( log(5) )}^{2} } + c$

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