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Question

# Find the indefinite integral. $\int\left(2+x+2 x^{2}+e^{x}\right) d x$

Solution

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Applying the extended version of $\textbf{Rule}$ $\pmb{4}$ we get:

$\int (2+x+2x^2+e^x)\, dx=\int 2\, dx+\int x\,dx+\int 2x^2\, dx+\int e^x\,dx$

Applying Rule $3$ on the first and the third integral we get

\begin{align*}\int (2+x+2x^2+e^x)\, dx&=\int 2\, dx+\int x\,dx+\int 2x^2\, dx+\int e^x\,dx\\ &=2\int \, dx+\int x\, dx+2\int x^2\, dx+\int e^x\, dx \end{align*}

Finally, applying Rule $5$ and Rule $2$ we have that

$\boxed{\int(2+x+2x^2+e^x)\, dx=2x+\frac{x^2}{2}+2\cdot \frac{x^3}{3}+e^x+C}$

Where $\textbf{Rule}$ $\pmb{4}$, $\textbf{Rule}$ $\pmb{5}$ , $\textbf{Rule}$ $\pmb{2}$ and $\textbf{Rule}$ $\pmb{3}$ are given by

$\pmb{\int [f(x)\pm g(x)]\, dx=\int f(x)\, dx \pm \int g(x)\, dx}$

$\pmb{\int e^x\, dx=e^x+C}$

$\pmb{\int x^n\, dx=\frac{1}{n+1}\cdot x^{n+1}+C}$

$\pmb{\int cf(x)\, dx=c\int f(x)\, dx}$

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