1. Some integration questions

1. $\int a^xe^xdx$
I tried to solve it by using integration by parts but couldnot get the required answer.

2. Originally Posted by roshanhero
1. $\int a^xe^xdx$
I tried to solve it by using integration by parts but couldnot get the required answer.
Try writing $a^x = e^{x \ln a}$

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

4. Originally Posted by roshanhero
1. $\int a^xe^xdx$
I tried to solve it by using integration by parts but couldnot get the required answer.
Let $y = (ae)^x$ then

$\ln y = x \ln (ae)$

$\frac{1}{y} \frac{dy}{dx} = \ln (ae)$

$dx = \frac{dy}{y \ln (ae)}$

and

$\int y ~dx = \int y \frac{dy}{y \ln (ae)}$

$\int y ~dx = \int \frac{dy}{\ln (ae)}$

5. How can we write $a^x = e^{x \ln a}$

6. Originally Posted by roshanhero
How can we write $a^x = e^{x \ln a}$
see Natural exponents and logarithms

7. Originally Posted by roshanhero
How can we write $a^x = e^{x \ln a}$
Its the definition of $a^x$ .

8. $\int a^xe^xdx=\int e^{xln a}e^x dx
=\int e^{x(log a+1)} dx$

I hope that upto now all is right. Now I can use the formula of integration of e^ax.

9. Originally Posted by roshanhero
$\int a^xe^xdx=\int e^{xln a}e^x dx
=\int e^{x(log a+1)} dx$

I hope that upto now all is right. Now I can use the formula of integration of e^ax.

10. I am sorry that I still cannot prove $a^x = e^{x \ln a}$ even with the link given by you guys.

11. Allright guys I proved it but in a messy way,tell me whether it is right or wrong
$
Let,a^x=y
$

$
ln a^x=lny
$

$
a^x=e^{lny}
$

$
a^x=e^{lna^x}
$

$
a^x=e^{xlna}
$

12. Originally Posted by roshanhero
Allright guys I proved it but in a messy way,tell me whether it is right or wrong
$
Let,a^x=y
$

$
ln a^x=lny
$

$
a^x=e^{lny}
$

$
a^x=e^{lna^x}
$

$
a^x=e^{xlna}
$
Yeh that's fine, it's probably less messy to just say

$e^{x \ln{a}} = \left( e^{\ln a} \right)^x = a^x$