What is the integral from 2 to 5 of t^2ln(3t) ?

The integral from 2 to 5 of t^2ln(3t) is -13+(ln((5^{125}*3^{117})/(256)))/3

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integral from 2 to 5 of t^2ln(3t)

\int_{2}^{5} t^{2} ln (3 t) d t

- 13 + \frac{ln ( \frac{5 ^{125} \cdot 3 ^{117}}{256} )}{3}

Decimal Notation

95.05739 \dots

Solution steps

\int_{2}^{5} t^{2} ln (3 t) d t

Apply Integration By Parts

: [\frac{1}{3} t^{3} ln (3 t) - \int \frac{t ^{2}}{3} d t]_{2}^{5}

= [\frac{1}{3} t^{3} ln (3 t) - \int \frac{t ^{2}}{3} d t]_{2}^{5}

\int \frac{t ^{2}}{3} d t = \frac{t ^{3}}{9}

= [\frac{1}{3} t^{3} ln (3 t) - \frac{t ^{3}}{9}]_{2}^{5}

Compute the boundaries:

- 13 + \frac{- 8 ln ( 6 ) + 125 ln ( 15 )}{3}

= - 13 + \frac{- 8 ln ( 6 ) + 125 ln ( 15 )}{3}

Simplify

= - 13 + \frac{ln ( \frac{5 ^{125} \cdot 3 ^{117}}{256} )}{3}

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What is the integral from 2 to 5 of t^2ln(3t) ?
The integral from 2 to 5 of t^2ln(3t) is -13+(ln((5^{125}*3^{117})/(256)))/3