What is the integral from 0 to l of (x/l)^2 ?

The integral from 0 to l of (x/l)^2 is l/3

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integral from 0 to l of (x/l)^2

\int_{0}^{l} (\frac{x}{l})^{2} d x

\frac{l}{3}

Solution steps

\int_{0}^{l} (\frac{x}{l})^{2} d x

Apply Integration By Parts

= [\frac{x ^{3}}{l ^{2}} - \int \frac{2 x ^{2}}{l ^{2}} d x]_{0}^{l}

\int \frac{2 x ^{2}}{l ^{2}} d x = \frac{2}{l ^{2}} \cdot \frac{x ^{3}}{3}

= [\frac{x ^{3}}{l ^{2}} - \frac{2}{l ^{2}} \cdot \frac{x ^{3}}{3}]_{0}^{l}

Join

\frac{x ^{3}}{l ^{2}} - \frac{2}{l ^{2}} \cdot \frac{x ^{3}}{3} : \frac{x ^{3}}{3 l ^{2}}

= [\frac{x ^{3}}{3 l ^{2}}]_{0}^{l}

Compute the boundaries:

\frac{l}{3}

= \frac{l}{3}

What is the integral from 0 to l of (x/l)^2 ?
The integral from 0 to l of (x/l)^2 is l/3