What is the integral from 0 to 12 of 1/3 ln(t+1) ?

The integral from 0 to 12 of 1/3 ln(t+1) is 1/3 (13ln(13)-12)

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integral from 0 to 12 of 1/3 ln(t+1)

\int_{0}^{12} \frac{1}{3} ln (t + 1) d t

\frac{1}{3} (13 ln (13) - 12)

Solution steps

\int_{0}^{12} \frac{1}{3} ln (t + 1) d t

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{3} \cdot \int_{0}^{12} ln (t + 1) d t

Apply u-substitution

: \int_{1}^{13} ln (u) d u

= \frac{1}{3} \cdot \int_{1}^{13} ln (u) d u

Apply Integration By Parts

: [u ln (u) - \int 1 d u]_{1}^{13}

= \frac{1}{3} [u ln (u) - \int 1 d u]_{1}^{13}

\int 1 d u = u

= \frac{1}{3} [u ln (u) - u]_{1}^{13}

Compute the boundaries:

13 ln (13) - 12

= \frac{1}{3} (13 ln (13) - 12)

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What is the integral from 0 to 12 of 1/3 ln(t+1) ?
The integral from 0 to 12 of 1/3 ln(t+1) is 1/3 (13ln(13)-12)