What is the integral from a to b of ln(r^2)r ?

The integral from a to b of ln(r^2)r is b^2ln(b)+(-b^2+a^2)/2-a^2ln(a)

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integral from a to b of ln(r^2)r

\int_{a}^{b} ln (r^{2}) r d r

b^{2} ln (b) + \frac{- b ^{2} + a ^{2}}{2} - a^{2} ln (a)

Solution steps

\int_{a}^{b} ln (r^{2}) r d r

Apply Integration By Parts

: [\frac{1}{2} r^{2} ln (r^{2}) - \int r d r]_{a}^{b}

= [\frac{1}{2} r^{2} ln (r^{2}) - \int r d r]_{a}^{b}

\int r d r = \frac{r ^{2}}{2}

= [\frac{1}{2} r^{2} ln (r^{2}) - \frac{r ^{2}}{2}]_{a}^{b}

\frac{1}{2} r^{2} ln (r^{2}) = r^{2} ln (r)

= [r^{2} ln (r) - \frac{r ^{2}}{2}]_{a}^{b}

Compute the boundaries:

b^{2} ln (b) - \frac{b ^{2}}{2} - a^{2} ln (a) + \frac{a ^{2}}{2}

= b^{2} ln (b) - \frac{b ^{2}}{2} - a^{2} ln (a) + \frac{a ^{2}}{2}

Simplify

= b^{2} ln (b) + \frac{- b ^{2} + a ^{2}}{2} - a^{2} ln (a)

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What is the integral from a to b of ln(r^2)r ?
The integral from a to b of ln(r^2)r is b^2ln(b)+(-b^2+a^2)/2-a^2ln(a)