What is the integral from 1/2 to e/2 of tln(2t) ?

The integral from 1/2 to e/2 of tln(2t) is (e^2+1)/(16)

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integral from 1/2 to e/2 of tln(2t)

\int_{\frac{1}{2}}^{\frac{e}{2}} t ln (2 t) d t

\frac{e ^{2} + 1}{16}

Decimal Notation

0.52431 \dots

Solution steps

\int_{\frac{1}{2}}^{\frac{e}{2}} t ln (2 t) d t

Apply Integration By Parts

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

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

\int \frac{t}{2} d t = \frac{t ^{2}}{4}

= [\frac{1}{2} (t^{2} ln (2 t) - 2 \cdot \frac{t ^{2}}{4})]_{\frac{1}{2}}^{\frac{e}{2}}

2 \cdot \frac{t ^{2}}{4} = \frac{t ^{2}}{2}

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

Compute the boundaries:

\frac{e ^{2}}{16} + \frac{1}{16}

= \frac{e ^{2}}{16} + \frac{1}{16}

Simplify

= \frac{e ^{2} + 1}{16}

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What is the integral from 1/2 to e/2 of tln(2t) ?
The integral from 1/2 to e/2 of tln(2t) is (e^2+1)/(16)