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Question Number 24363 by gopikrishnan005@gmail.com last updated on 16/Nov/17

∫(ae)^x dx

$$\int\left({ae}\right)^{{x}} {dx} \\ $$

Answered by ajfour last updated on 16/Nov/17

=(((ae)^x )/(1+ln a))+C .

$$=\frac{\left({ae}\right)^{{x}} }{\mathrm{1}+\mathrm{ln}\:{a}}+{C}\:. \\ $$

Commented by gopikrishnan005@gmail.com last updated on 16/Nov/17

pls explain

$${pls}\:{explain} \\ $$

Answered by abwayh last updated on 16/Nov/17

let u=(ae)^x ln u=xln (ae) ln u=x(ln a+1) (1/u) du=(ln a+1)dx dx=(du/(u(ln a+1))) ∫(ae)^x dx=∫(( udu)/(u(ln a+1)))=∫(du/((ln a+1)))=(u/((ln a+1)))+c =(((ae)^x )/((lna+1)))+c

$$\mathrm{let}\:\:\mathrm{u}=\left(\mathrm{ae}\right)^{\mathrm{x}} \:\: \\ $$$$\mathrm{ln}\:\mathrm{u}=\mathrm{xln}\:\left(\mathrm{ae}\right) \\ $$$$\mathrm{ln}\:\mathrm{u}=\mathrm{x}\left(\mathrm{ln}\:\mathrm{a}+\mathrm{1}\right) \\ $$$$\frac{\mathrm{1}}{\mathrm{u}}\:\mathrm{du}=\left(\mathrm{ln}\:\mathrm{a}+\mathrm{1}\right)\mathrm{dx} \\ $$$$\mathrm{dx}=\frac{\mathrm{du}}{\mathrm{u}\left(\mathrm{ln}\:\mathrm{a}+\mathrm{1}\right)} \\ $$$$\int\left(\mathrm{ae}\right)^{\mathrm{x}} \mathrm{dx}=\int\frac{\:\:\mathrm{udu}}{\mathrm{u}\left(\mathrm{ln}\:\mathrm{a}+\mathrm{1}\right)}=\int\frac{\mathrm{du}}{\left(\mathrm{ln}\:\mathrm{a}+\mathrm{1}\right)}=\frac{\mathrm{u}}{\left(\mathrm{ln}\:\mathrm{a}+\mathrm{1}\right)}+\mathrm{c} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\frac{\left(\mathrm{ae}\right)^{\mathrm{x}} }{\left(\mathrm{lna}+\mathrm{1}\right)}+\mathrm{c} \\ $$

Commented by gopikrishnan005@gmail.com last updated on 17/Nov/17

thank u sir..one dbt loge value 1 ah sir

$${thank}\:{u}\:{sir}..{one}\:{dbt}\:{loge}\:{value}\:\mathrm{1}\:{ah}\:{sir} \\ $$

Answered by A1B1C1D1 last updated on 17/Nov/17

There integral must be done piecewtemose: For the interval where: log (ae) = 0 ∫ (ea)^x dx = (((ea)^x )/(ln(ea))) our x for log (a + 1) = 0 The answere is: { ((x for log (a + 1) = 0)),(((((ea)^x )/(ln(ea))) otherwtemose ^( + C) )) :}

$$ \\ $$$$ \\ $$$$\mathrm{There}\:\mathrm{integral}\:\mathrm{must}\:\mathrm{be}\:\mathrm{done}\:\mathrm{piecewtemose}: \\ $$$$ \\ $$$$\mathrm{For}\:\mathrm{the}\:\mathrm{interval}\:\mathrm{where}: \\ $$$$ \\ $$$$\mathrm{log}\:\left(\mathrm{ae}\right)\:=\:\mathrm{0} \\ $$$$ \\ $$$$\int\:\left(\mathrm{ea}\right)^{\mathrm{x}} \:\mathrm{dx}\:=\:\frac{\left(\mathrm{ea}\right)^{\mathrm{x}} }{\mathrm{ln}\left(\mathrm{ea}\right)}\:\mathrm{our}\:\mathrm{x}\:\mathrm{for}\:\mathrm{log}\:\left(\mathrm{a}\:+\:\mathrm{1}\right)\:=\:\mathrm{0} \\ $$$$ \\ $$$$\mathrm{The}\:\mathrm{answere}\:\mathrm{is}: \\ $$$$ \\ $$$$\begin{cases}{\mathrm{x}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{for}\:\mathrm{log}\:\left(\mathrm{a}\:+\:\mathrm{1}\right)\:=\:\mathrm{0}}\\{\frac{\left(\mathrm{ea}\right)^{\mathrm{x}} }{\mathrm{ln}\left(\mathrm{ea}\right)}\:\:\:\:\:\mathrm{otherwtemose}\:\:\:\:\:\:\:\overset{\:\:\:\:\:\:\:+\:\mathrm{C}} {\:}}\end{cases} \\ $$$$ \\ $$

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