Question Number 24363 by gopikrishnan005@gmail.com last updated on 16/Nov/17
$$\int\left({ae}\right)^{{x}} {dx} \\ $$
Answered by ajfour last updated on 16/Nov/17
$$=\frac{\left({ae}\right)^{{x}} }{\mathrm{1}+\mathrm{ln}\:{a}}+{C}\:. \\ $$
Commented by gopikrishnan005@gmail.com last updated on 16/Nov/17
$${pls}\:{explain} \\ $$
Answered by abwayh last updated on 16/Nov/17
$$\mathrm{let}\:\:\mathrm{u}=\left(\mathrm{ae}\right)^{\mathrm{x}} \:\: \\ $$$$\mathrm{ln}\:\mathrm{u}=\mathrm{xln}\:\left(\mathrm{ae}\right) \\ $$$$\mathrm{ln}\:\mathrm{u}=\mathrm{x}\left(\mathrm{ln}\:\mathrm{a}+\mathrm{1}\right) \\ $$$$\frac{\mathrm{1}}{\mathrm{u}}\:\mathrm{du}=\left(\mathrm{ln}\:\mathrm{a}+\mathrm{1}\right)\mathrm{dx} \\ $$$$\mathrm{dx}=\frac{\mathrm{du}}{\mathrm{u}\left(\mathrm{ln}\:\mathrm{a}+\mathrm{1}\right)} \\ $$$$\int\left(\mathrm{ae}\right)^{\mathrm{x}} \mathrm{dx}=\int\frac{\:\:\mathrm{udu}}{\mathrm{u}\left(\mathrm{ln}\:\mathrm{a}+\mathrm{1}\right)}=\int\frac{\mathrm{du}}{\left(\mathrm{ln}\:\mathrm{a}+\mathrm{1}\right)}=\frac{\mathrm{u}}{\left(\mathrm{ln}\:\mathrm{a}+\mathrm{1}\right)}+\mathrm{c} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\frac{\left(\mathrm{ae}\right)^{\mathrm{x}} }{\left(\mathrm{lna}+\mathrm{1}\right)}+\mathrm{c} \\ $$
Commented by gopikrishnan005@gmail.com last updated on 17/Nov/17
$${thank}\:{u}\:{sir}..{one}\:{dbt}\:{loge}\:{value}\:\mathrm{1}\:{ah}\:{sir} \\ $$
Answered by A1B1C1D1 last updated on 17/Nov/17
$$ \\ $$$$ \\ $$$$\mathrm{There}\:\mathrm{integral}\:\mathrm{must}\:\mathrm{be}\:\mathrm{done}\:\mathrm{piecewtemose}: \\ $$$$ \\ $$$$\mathrm{For}\:\mathrm{the}\:\mathrm{interval}\:\mathrm{where}: \\ $$$$ \\ $$$$\mathrm{log}\:\left(\mathrm{ae}\right)\:=\:\mathrm{0} \\ $$$$ \\ $$$$\int\:\left(\mathrm{ea}\right)^{\mathrm{x}} \:\mathrm{dx}\:=\:\frac{\left(\mathrm{ea}\right)^{\mathrm{x}} }{\mathrm{ln}\left(\mathrm{ea}\right)}\:\mathrm{our}\:\mathrm{x}\:\mathrm{for}\:\mathrm{log}\:\left(\mathrm{a}\:+\:\mathrm{1}\right)\:=\:\mathrm{0} \\ $$$$ \\ $$$$\mathrm{The}\:\mathrm{answere}\:\mathrm{is}: \\ $$$$ \\ $$$$\begin{cases}{\mathrm{x}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{for}\:\mathrm{log}\:\left(\mathrm{a}\:+\:\mathrm{1}\right)\:=\:\mathrm{0}}\\{\frac{\left(\mathrm{ea}\right)^{\mathrm{x}} }{\mathrm{ln}\left(\mathrm{ea}\right)}\:\:\:\:\:\mathrm{otherwtemose}\:\:\:\:\:\:\:\overset{\:\:\:\:\:\:\:+\:\mathrm{C}} {\:}}\end{cases} \\ $$$$ \\ $$