Question

What is \[\int {{a^x}{e^x}\,dx} \]equal to?
A. $ \dfrac{{{a^x}{e^x}}}{{\ln \,a}} + \,c\, $ where c is the constant of integration
B. $ {a^x}{e^x} $ + c where c is the constant of integration
C. $ \dfrac{{{a^x}{e^x}}}{{\ln (ae)}} $ + c where c is the constant of integration
D. None of the above

Answer 1

Hint: n this question, we can clearly see that \[\int {{a^x}{e^x}\,dx} \] is in the form of $ \int {u.v\,dx} $ . So, we can use the formula of integration of two functions or integration by parts. Choose first and second function according to the ILATE rule. Here $ {a^x} $ will be the first function and $ {e^x} $ will be the second function. Proceed with simplification after applying the product formula.

Complete step-by-step answer:
Let I = \[\int {{a^x}{e^x}\,dx} \]
We know that, $ \int {u.vdx = u\int {vdx\, - \,\int {\left[ {\dfrac{d}{{dx}}u.\int {vdx} } \right]} } } \,dx $
Where u = $ {a^x} $ and v = $ {e^x} $ .
 $ I $ = $ \int {{a^x}} .{e^x}\,dx\, = \,{a^x}.\int {{e^x}} dx\, - \,\int {\left[ {\dfrac{d}{{dx}}{a^x}.\,\int {{e^x}} dx} \right]} dx $ (Putting the values of u and v in the above formula)
 $\Rightarrow I $ = $ {a^x}.{e^x} - \,\int {{a^x}} .\ln a\,.\,{e^x}\,dx $ (We know that $ \int {{e^x}} = {e^x} $ and $ \dfrac{d}{{dx}}{a^x} = {a^x}.\ln a $ )
 $\Rightarrow I $ = $ {a^x}.{e^x}\, - \,\ln \,a\int {{a^x}} .{e^x}dx $ (ln a is constant)
 $\Rightarrow I $ = $ {a^x}.{e^x} - \,I\ln a $ + c (Putting the value of \[\int {{a^x}{e^x}\,dx} \]= I)
 $ I + I\,\ln \,a\, = \,{a^x}.{e^x} $ + c
 $ I\left( {1 + \ln a} \right) = \,{a^x}{e^x} $ + c
 $\Rightarrow I = \dfrac{{{a^x}{e^x}}}{{1 + \ln \,a}} $ + c
 $\Rightarrow I = \dfrac{{{a^x}{e^x}}}{{\ln \,e\, + \,\ln a}} + c $ (We know that the value of ln e = 1)
 $\Rightarrow I = \dfrac{{{a^x}{e^x}}}{{\ln ae}} + c $

So, the correct answer is “Option C”.

Note: Integration by parts formula $ \int {u.vdx = u\int {vdx\, - \,\int {\left[ {\dfrac{d}{{dx}}u.\int {vdx} } \right]} } } \,dx $ . This formula is used for integrating the product of two functions. But you should note that this formula is not applicable for functions such as $ \int {\sqrt x } \cos xdx $ . But this formula will be applicable on $ \int {x\cos x} $ . One more thing you need to keep in mind that we do not add any constant while finding the integral of the second function.
There is a rule of choosing the first function and the second function, that rule is called ILATE. I stands for inverse trigonometric function, L stands for logarithmic function, A stands for algebraic function, T stands for trigonometric function and E stands for exponential function. Always follow this rule while integrating by product. If you do the integration randomly, the solution becomes much more complicated.