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Question
\[\int\sqrt{1 + e^x} . e^x dx\]
Sum
Solution
\[\int\sqrt{1 + e^x} \cdot e^x dx\]
\[\text{Let 1 }+ e^x = t\]
\[ \Rightarrow e^x = \frac{dt}{dx}\]
\[ \Rightarrow e^x dx = dt\]
\[Now, \int\sqrt{1 + e^x} \cdot e^x dx\]
\[ = \int\sqrt{t} \cdot dt\]
\[ = \frac{t^\frac{1}{2} + 1}{\frac{1}{2} + 1} + C\]
\[ = \frac{2}{3} t^\frac{3}{2} + C\]
\[ = \frac{2}{3} \left( 1 + e^x \right)^\frac{3}{2} + C\]
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