Question

Give Me Correct ✅ Answer Please
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Answer 1

Question:-5

$\displaystyle \rm \: \text{Evaluate \: :-} \int 5^{5^{x}} \cdot 5^{x} d x .$

Solution:-

$\text{Given: \( \displaystyle \int \rm 5^{5^{x}} \cdot 5^{x} \cdot d x \)}$

To Find: Evaluate

Solution:

$\text{ \( \displaystyle \rm I=\int 5^{5^{x}} \cdot 5^{x} \cdot d x \) put \(\displaystyle \rm 5^{x}=t \)}$

$\begin{array}{}\displaystyle \rm \therefore 5^{x} \cdot \log 5 \cdot d x=1 d t \\ \\ \displaystyle \rm 5^{x} \cdot d x=\frac{1}{\log 5} \cdot d t \\ \\ \displaystyle \rm=\int 5^{t} \cdot \frac{1}{\log 5} \cdot d t \\ \\ \displaystyle \rm I=\frac{1}{\log 5} \cdot \int 5^{t} \cdot d t \\ \\ \displaystyle \rm =\frac{1}{\log 5} \cdot 5 t \cdot \frac{1}{\log 5}+c \\ \\ \displaystyle \rm =\left(\frac{1}{\log 5}\right)^{2} \cdot 5^{ 5^{x}}+c \end{array}$

Hence, answer is $\displaystyle \rm \red{\left(\frac{1}{\log 5}\right)^{2} \cdot 5 ^{5^{x}}+c . }$