# Evaluate the integral displaystyleintfrac{x}{{2}}{left.{d}{x}right.}

Question
Integrals
Evaluate the integral $$\displaystyle\int\frac{x}{{2}}{\left.{d}{x}\right.}$$

2021-02-10
There are two properties of integrals that we'll use to evaluate this.
1.Constant multiples can be pulled out of the integral.
$$\displaystyle\int{k} f{{\left({x}\right)}}{\left.{d}{x}\right.}={k}\int f{{\left({x}\right)}}{\left.{d}{x}\right.}$$
2.The integral of $$\displaystyle{x}^{n}$$ is given by
$$\displaystyle\int{x}^{n}{\left.{d}{x}\right.}=\frac{{{x}^{{{n}+{1}}}}}{{{n}+{1}}}+{C}$$
which you can easily show using the Fundamental Theorem of Calculus, if you've gotten to that yet.
Using these two properties, we thus see
$$\displaystyle\int\frac{x}{{2}}{\left.{d}{x}\right.}=\int\frac{1}{{2}}{x}^{1}{\left.{d}{x}\right.}$$
$$\displaystyle=\frac{1}{{2}}\int{x}^{1}{\left.{d}{x}\right.}$$
$$\displaystyle=\frac{1}{{2}}\frac{{x}^{2}}{{2}}+{C}$$
$$\displaystyle=\frac{1}{{4}}{x}^{2}+{C}$$

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