What is the integral from 0 to 4 of ln(3x+1) ?

The integral from 0 to 4 of ln(3x+1) is 1/3 (13ln(13)-12)

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integral from 0 to 4 of ln(3x+1)

\int_{0}^{4} ln (3 x + 1) d x

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

Decimal

7.11478 \dots

Solution steps

\int_{0}^{4} ln (3 x + 1) d x

Apply u-substitution

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

Take the constant out:

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

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

Apply Integration By Parts

= \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)

What is the integral from 0 to 4 of ln(3x+1) ?
The integral from 0 to 4 of ln(3x+1) is 1/3 (13ln(13)-12)