What is the integral from-3 to 0 of ln(y^2+5) ?

The integral from-3 to 0 of ln(y^2+5) is -6+3ln(14)+2sqrt(5)arctan(3/(sqrt(5)))

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integral from-3 to 0 of ln(y^2+5)

\int_{- 3}^{0} ln (y^{2} + 5) d y

- 6 + 3 ln (14) + 25 arctan (\frac{3}{5})

Decimal Notation

6.07748 \dots

Solution steps

\int_{- 3}^{0} ln (y^{2} + 5) d y

Apply Integration By Parts

: [y ln (y^{2} + 5) - \int \frac{2 y ^{2}}{y ^{2} + 5} d y]_{- 3}^{0}

= [y ln (y^{2} + 5) - \int \frac{2 y ^{2}}{y ^{2} + 5} d y]_{- 3}^{0}

\int \frac{2 y ^{2}}{y ^{2} + 5} d y = - 25 arctan (\frac{y}{5}) + 2 y

= [y ln (y^{2} + 5) - (- 25 arctan (\frac{y}{5}) + 2 y)]_{- 3}^{0}

Simplify

= [y ln (y^{2} + 5) + 25 arctan (\frac{y}{5}) - 2 y]_{- 3}^{0}

Compute the boundaries:

- 6 + 3 ln (14) + 25 arctan (\frac{3}{5})

= - 6 + 3 ln (14) + 25 arctan (\frac{3}{5})

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What is the integral from-3 to 0 of ln(y^2+5) ?
The integral from-3 to 0 of ln(y^2+5) is -6+3ln(14)+2sqrt(5)arctan(3/(sqrt(5)))