Evaluate the indefinite integral. {eq}\int e^x \sqrt{1 + e^x} \, \mathrm{d}x {/eq}

Question:

Evaluate the indefinite integral.

{eq}\int e^x \sqrt{1 + e^x} \, \mathrm{d}x {/eq}

Substitution Method of Integration:

In this method, we need to set u equal to the function, differentiate it, and rewrite the integral in terms of a new variable. In the end, apply the integral formula and properties to solve the integral problem. This method is also called the change of variable method of integration. It can change the form of the integrand and makes it preferable to one that we can integrate more easily.

Some integral formulas are given below:

{eq}\begin{align*} \int {{x^n}dx} &= \dfrac{{{x^{n + 1}}}}{{n + 1}} + c\\ \int {1dx} &= x + c\\ \int {\cos x} &= \sin x + c\\ \int {{e^x}dx} &= {e^x} + c \end{align*} {/eq}

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Given Data:

• The given function is: {eq}I = \int {{e^x}\sqrt {1 + {e^x}} dx} {/eq}.

Apply u-substitution: {eq}1 + {e^x} = t {/eq}, and...