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Question

# Find the indefinite integral. $\int\left(1+u+u^{2}\right) d u$

Solution

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Applying $\textbf{Rule}$,,$\pmb{4}$ we get

\begin{align*} \displaystyle\int{(1+u+u^2)\, du}&= \displaystyle\int{\, du}+\displaystyle\int{u\, du}+\displaystyle\int{u^2\, du}\tag{\textcolor{#c34632}{Rule 2}}\\ &=u+\frac{u^{1+1}}{1+1}+\frac{u^{2+1}}{2+1}+C\\ \end{align*}

Finally, we get

$\boxed{\boxed{\displaystyle\int{(1+u+u^2)\, du=u+\frac{u^2}{2}+\frac{u^3}{3}+C}}}$

Where $\textbf{Rule $2$}$ and $\textbf{Rule $4$}$ are given by

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

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

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