Question

Determine the $\int {{5^{{5^{{5^x}}}}}} {.5^{{5^x}}}{.5^x}dx$

Answer 1

Hint: According to given in the question we have to determine the value if the integration$\int {{5^{{5^{{5^x}}}}}} {.5^{{5^x}}}{.5^x}dx$. So, first of all we have to let the ${5^{{5^x}}}$as any of the integer than we have to determine the differentiation we just let and to determine the differentiation we have to apply the formula as mentioned below:

Formula used:
$ \Rightarrow \int {{a^x}dx = {a^x}\log _e^{}a.............(A)} $
Now, we have to substitute the obtained differentiation in the integration as given in the question to determine the value of the integration.
Now, we have to find the integration of the expression just obtained by taking the constant terms separate.

$ \Rightarrow \int {{a^x}dx = \dfrac{{{a^x}}}{{\log a}}.............(B)} $
After that now, we have to substitute the value we let in the integration we solved to determine the final answer or we can say that the value of the integration given in the question.

Complete step by step answer:
Step 1: First of all we have to let the ${5^{{5^x}}}$ as any of the integers as mentioned in the solution hint. Hence,
$ \Rightarrow {5^{{5^x}}} = t........(1)$
Step 2: Now, we have to determine the differentiation of the expression (1) as obtained in the solution step 1 with the help of the formula (A) as mentioned in the solution hint. Hence,
$
   \Rightarrow \dfrac{{d{5^{{5^x}}}}}{{dx}} = \dfrac{{dt}}{{dx}} \\
   \Rightarrow {5^{{5^x}}}\log 5x{5^x}\log 5 = \dfrac{{dt}}{{dx}} \\
   \Rightarrow {5^{{5^x}}}{5^x}dx = \dfrac{{dt}}{{{{(\log 5)}^2}}} \\
 $
Step 3: Now, we have to substitute the obtained differentiation as obtained in the solution step 2 in the given integration. Hence,
$ \Rightarrow I = \int {{5^t}} \dfrac{{dt}}{{{{(\log 5)}^2}}}$………(2)
Step 4: Now, from the expression (2) as obtained in the solution step 3 we have to take constant terms separately to determine the integration of the terms left as explained in the solution hint.
$ \Rightarrow I = \dfrac{1}{{{{(\log 5)}^2}}}\int {{5^t}} dt...................(3)$
Step 5: Now, to find the integration of the expression (3) as obtained in the solution step 4 we have to use the formula (B) as mentioned in the solution hint.
$ \Rightarrow I = \dfrac{1}{{{{(\log 5)}^2}}}\dfrac{{{5^t}}}{{\log 5}}$
Now, on substituting the value of t as we let in the solution step 1 hence,
$
   \Rightarrow I = \dfrac{1}{{{{(\log 5)}^2}}}\dfrac{{{5^{{5^{{5^x}}}}}}}{{\log 5}} + c \\
   \Rightarrow I = \dfrac{{{5^{{5^{{5^x}}}}}}}{{{{(\log 5)}^3}}} + c \\
 $

Hence, with the help of the formulas (A) and (B) we have determine the value of the given integration which is $\int {{5^{{5^{{5^x}}}}}} {.5^{{5^x}}}{.5^x}dx = \dfrac{{{5^{{5^{{5^x}}}}}}}{{{{(\log 5)}^3}}} + c$

Note: To find the integration as given in the question it is necessary that we have to let ${5^{{5^x}}}$ as any of the integer can be x, y, z, or t so, that we can easily solve the integration but then we have to determine the differentiation of the term we let.