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Find the value of the given integral
\[\int {{a^x}{e^x}dx} \]

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Hint:- Use the integral by-parts.

Let the value of the given integral be I.
Then, I = \[\int {{a^x}{e^x}dx} \]. (1)
As, we know that if u and v are two functions of $x$ , then the integral of the product of
these two functions will be:
\[ \Rightarrow \int {uvdx = u\int {vdx - \int {\left[ {\dfrac{{du}}{{dx}}\int {vdx} } \right]dx} } } \] (2)
In applying the above equation, the selection of the first function (u) and
Second function (v) should be done depending on which function can be integrated easily.
Normally, we use the preference order for the first function i.e.
ILATE RULE (Inverse, Logarithmic, Algebraic, Trigonometric, Exponent) which states that the
Inverse function should be assumed as the first function while performing the integration.
Hence the functions are assumed from left to right depending on the type of functions involved.
Then by using the ILATE Rule. We can easily solve the above problem.
According to ILATE Rule,
\[ \Rightarrow u = {a^x}\]
\[ \Rightarrow v = {e^x}\]
So, now putting value of u and v in equation 2 we get,
\[ \Rightarrow I = \int {{a^x}{e^x}dx = {a^x}\int {{e^x}dx - \int {\left[ {\dfrac{{d\left( {{a^x}} \right)}}{{dx}}\int {{e^x}dx} } \right]dx} } } \] (3)
As, we know that, \[\int {{e^x}dx = {e^x}} \]and \[\dfrac{{d\left( {{a^x}} \right)}}{{dx}} = {a^x}.\ln a\]
So, now solving equation 3 we get,
\[ \Rightarrow I = {a^x}.{e^x} - \ln a\int {{a^x}.{e^x}dx} \]
Now, putting the value of \[\int {{a^x}{e^x}dx} \] from equation 1 to above equation. We get,
\[ \Rightarrow I = {a^x}.{e^x} - \ln a(I)\]
Solving above equation we get,
\[
   \Rightarrow I\left( {1 + \ln a} \right) = {a^x}.{e^x} \\
   \Rightarrow I = \dfrac{{{a^x}.{e^x}}}{{\left( {1 + \ln a} \right)}} \\
 \]
Hence the value of given integral is \[\int {{a^x}{e^x}dx} = \dfrac{{{a^x}.{e^x}}}{{\left( {1 + \ln a} \right)}}\].

NOTE:- Whenever we came up with this type of problem then easiest and efficient way to
Solving the problem is using by-parts. And for the selection of the first function we can use ILATE
RULE.Then we can find the value of the given integral using parts. But remember the basic
differentiation and integration formulas.

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