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$\int e^{(e^{x})}e^{x}dx$

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Sin, cos, and tan of special angles

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Q.

\int e^{(e^{x})}e^{x}dx

Recognize Pattern: Recognize the pattern in the integral. $\newline$ We are given the integral of the function $e^{e^{x}} \cdot e^{x}$ with respect to $x$ . This resembles the pattern of an exponential function where the derivative of the exponent is present outside the exponential function.
Make Substitution: Let $u = e^{x}$ . $\newline$ We make a substitution to simplify the integral. Let $u = e^{x}$ . Then, we need to find the derivative of $u$ with respect to $x$ , which is $\frac{du}{dx} = e^{x}$ .
Rewrite in Terms of $u$ : Rewrite the integral in terms of $u$ . Since $\frac{du}{dx} = e^{x}$ , we can write $dx = \frac{du}{e^{x}}$ . Now, we can substitute $u$ for $e^{x}$ and $\frac{du}{e^{x}}$ for $dx$ in the integral. The integral becomes $\int e^{u} \cdot \left(\frac{du}{e^{x}}\right)$ .
Simplify Integral: Simplify the integral. $\newline$ Notice that $e^{x}$ is $u$ , so we can cancel $e^{x}$ in the numerator and the denominator. The integral simplifies to $\int e^{u} \, du$ .
Integrate with $u$ : Integrate with respect to $u$ . The integral of $e^{u}$ with respect to $u$ is $e^{u} + C$ , where $C$ is the constant of integration.
Substitute Back: Substitute back in terms of $x$ . We originally let $u = e^{x}$ , so we substitute back to get the integral in terms of $x$ . The result is $e^{e^{x}} + C$ .

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