Evaluate the indefinite integral. integral e^x (e^x + 3)^{12} dx.

Question:

Evaluate the indefinite integral.

{eq}\displaystyle \int e^x (e^x + 3)^{12}\ dx {/eq}.

Indefinite Integral in Calculus:

In this problem, We need to compute the given an indefinite integral. There are many different ways to do it. Integration by substitution is one of the major ways, which is a process of temporary substituting the given variables by a new variable.

Next, we'll apply integral power rule to get the desired solution.

We are given:

{eq}\displaystyle \int e^x (e^x + 3)^{12} \ dx {/eq}

Apply u-substitution {eq}\displaystyle u= e^{x}+3 \rightarrow \ du = e^{x} \ dx {/eq}

{eq}= \displaystyle \int u^{12} \ du {/eq}

Apply integral power rule:

{eq}= \displaystyle \dfrac{ u^{12+1}}{12+1}+C {/eq}

{eq}= \displaystyle \dfrac{ u^{13}}{13}+C {/eq}

Substitute back {eq}\displaystyle u=e^{x}+3 {/eq}

{eq}= \displaystyle \dfrac{ (e^x+3)^{13}}{13}+C {/eq}

Therefore, the solution is {eq}\displaystyle {\boxed{ \int e^x (e^x + 3)^{12} \ dx = \displaystyle \dfrac{ (e^x+3)^{13}}{13}+C }} {/eq}