Answer
Verified
91.2k+ views
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.
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.
Recently Updated Pages
Name the scale on which the destructive energy of an class 11 physics JEE_Main
Write an article on the need and importance of sports class 10 english JEE_Main
Choose the exact meaning of the given idiomphrase The class 9 english JEE_Main
Choose the one which best expresses the meaning of class 9 english JEE_Main
What does a hydrometer consist of A A cylindrical stem class 9 physics JEE_Main
A motorcyclist of mass m is to negotiate a curve of class 9 physics JEE_Main
Other Pages
Electric field due to uniformly charged sphere class 12 physics JEE_Main
The vapour pressure of pure A is 10 torr and at the class 12 chemistry JEE_Main
3 mole of gas X and 2 moles of gas Y enters from the class 11 physics JEE_Main
If the distance between 1st crest and the third crest class 11 physics JEE_Main
A man of mass 50kg is standing on a 100kg plank kept class 11 physics JEE_Main
If a wire of resistance R is stretched to double of class 12 physics JEE_Main