1.$\displaystyle \int a^xe^xdx$
I tried to solve it by using integration by parts but couldnot get the required answer.
Assuming $\displaystyle a>0$
Actually it must be positive since the base must be positive.
$\displaystyle \int a^{x} e^{x} dx = C1 + a^{x} e^{x} - ln(a) \int a^{x} e^{x} dx $
then
$\displaystyle (1+ln(a)) \int a^{x} e^{x} dx = a^{x} e^{x} + C1$
$\displaystyle \int a^{x} e^{x} dx = \frac{a^{x} e^{x}}{1+ln(a)} + C$
where $\displaystyle C = \frac{C1}{1+ln(a)} $
Let $\displaystyle y = (ae)^x$ then
$\displaystyle \ln y = x \ln (ae)$
$\displaystyle \frac{1}{y} \frac{dy}{dx} = \ln (ae)$
$\displaystyle dx = \frac{dy}{y \ln (ae)} $
and
$\displaystyle \int y ~dx = \int y \frac{dy}{y \ln (ae)}$
$\displaystyle \int y ~dx = \int \frac{dy}{\ln (ae)}$