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

Question
Integrals
asked 2021-02-09
Evaluate the integral \(\displaystyle\int\frac{x}{{2}}{\left.{d}{x}\right.}\)

Answers (1)

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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