What is the integral from 0 to 2 of ln(3-x) ?

The integral from 0 to 2 of ln(3-x) is 3ln(3)-2

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integral from 0 to 2 of ln(3-x)

\int_{0}^{2} ln (3 - x) d x

3 ln (3) - 2

Decimal

1.29583 \dots

Solution steps

\int_{0}^{2} ln (3 - x) d x

Apply u-substitution

: \int_{3}^{1} - ln (u) d u

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

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

\int 1 d u = u

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

Simplify

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

Compute the boundaries:

3 ln (3) - 2

= 3 ln (3) - 2

\int_{- 1}^{1} x arccos (x) d x

\int_{0}^{1} \frac{x}{( 1 - x )} d x

\int_{0}^{2} x^{2} - 40 d x

\int_{1}^{3} x^{2} - 3 x + 1 d x

\int_{3}^{4} (3 x^{2} - 1) d x

What is the integral from 0 to 2 of ln(3-x) ?
The integral from 0 to 2 of ln(3-x) is 3ln(3)-2