What is the integral from 0 to 2 of ln(x^2+1) ?

The integral from 0 to 2 of ln(x^2+1) is 2ln(5)-2(-arctan(2)+2)

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integral from 0 to 2 of ln(x^2+1)

\int_{0}^{2} ln (x^{2} + 1) d x

2 ln (5) - 2 (- arctan (2) + 2)

Decimal

1.43317 \dots

Solution steps

\int_{0}^{2} ln (x^{2} + 1) d x

Apply Integration By Parts

= [x ln (x^{2} + 1) - \int \frac{2 x ^{2}}{x ^{2} + 1} d x]_{0}^{2}

\int \frac{2 x ^{2}}{x ^{2} + 1} d x = 2 (- arctan (x) + x)

= [x ln (x^{2} + 1) - 2 (- arctan (x) + x)]_{0}^{2}

Compute the boundaries:

2 ln (5) - 2 (- arctan (2) + 2)

= 2 ln (5) - 2 (- arctan (2) + 2)

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\int_{0}^{3} x^{2} e^{- 3 x} d x

\int_{r}^{\infty} \frac{1}{r ^{2}} d r

\int_{1}^{5} ∣ x - 2 ∣ d x

\int_{0}^{3} 9 x^{2} - x^{4} d x

What is the integral from 0 to 2 of ln(x^2+1) ?
The integral from 0 to 2 of ln(x^2+1) is 2ln(5)-2(-arctan(2)+2)